functions.wolfram.com

Wolfram Research’s Mathematical Functions;
Providing the mathematical and scientific community 
with the single most comprehensive and unified resourse 
of information about mathematical functions
(currently more than 35000 formulas for over 250 functions)

Oleg Marichev and Michael Trott

Abstract

Currently the website www.functions.wolfram.com includes more than 37000 formulas for the about 250 elementary and special functions that are available in Mathematica. 
The structure and organization of the website is discussed in detail. On overview is given how to manage and update this large mathematical knowledge base and how to deliver various formats types (Mathematica notebooks, html, and pdf).
Various "semiautomatic" ways to find new formulas using Mathematica will be demonstrated.
Future development directions will be outlined.

Why make such a website?

Existing books are no longer appropriate

Abramowitz/Stegun and Bateman/Erdelyi

★ Abramowitz/Stegun (and Bateman/Erdelyi) are "old" (by year 2001 standards)—many new results have been found since 1964 (1950-54).

★ Abramowitz/Stegun isn't comprehensive; missing are PolyLog, MeijerG, ….

★ There is no longer a need for tables of numerical values, thanks to inexpensive computing power and the wide availability of high–precision numeric capabilities for all complex values of all special functions.

★ Abramowitz/Stegun is not uniform; its various chapters were compiled by different authors with different criteria for inclusion of material.

★ Often there is no closed form for a function value, but there is an algorithm that produces as many terms as one wants.
   (Example: series expansions for zeros of Bessel functions \(TraditionalForm\`ξ\_n\%\((ν)\) = ∑\+\(k = 0\)\%∞\( c\_k\) \(f(n)\)\^\(-k\)\).)

★ A lot of classical formulas for functions with branch cuts are not developed appropriately on branch cuts.

Existing formulas should be collected in one place

Our goal is to collect all available material about special functions on this site.

Necessity to revise existing and add new formulas and information

Building this site will start with the over 250 built–in functions of Mathematica and then we will cover additional special functions. (For these additional special functions we will also maintain a Mathematica code repository).

To provide maximal modern technological support

The site is based on Mathematica notebook technology that enables the automatic generation of different presentation media. Current formats include HTML, PDF, and Mathematica notebooks. Future formats will include XML/MathML as it becomes more prevalent.

To provide permanent updating

Contributions from researchers are welcome to help expand the scope of the site.

To provide abilities for using computer technology in mathematics and applications

There is a real need for interactive an electronic version, allowing the user to manipulate integral representations, exchange integration and summation, ....

Resources for making such  website

Excellent team

Conception and support

Stephen Wolfram conceived this site and directed its development.

Mathematical group

Oleg Marichev, Michael Trott, Peter Falloon

Formatting

Andy Hunt

Design

Jeremy Davis, Kenny Kim

Web site

Jean Buck, Evaldas Imbrasas

Former participants

David Gehrig, Michael Chiong

Library

* 1000 checked out books
* 2000 copied articles (and www.arxive.org of course).

Existing and permanently improving system Mathematica

General remarks

Mathematica already has extraordinary capabilities in calculation, plotting, programming, publication, data manipulation, etc. New functionalities and functions are always being added to this already very general system. And this is true not only of the world of mathematics, but also other uses of the computer, such as electronic publishing by Mathematica-generated web site.

Mathematica supports over 250 elementary and special functions (more than exist in any other systems).

TableForm[
      Select[Names["*"], 
        Function[f, MemberQ[Attributes[f], NumericFunction]]]]

These functions have  "natural" naming (for example, LegendreP, HypergeometricPFQ, …). They can be:
* manipulated algebraically
* differentiated
* integrated
* plotted
* numerical values can be calculated to arbitrary precision

Section 3.2.10,  Section 3.2.11,  Section 3.2.12

Branch cuts are more consistent in Mathematica than in mathematics itself!

All branch cuts follow uniquely from the branch cut of Log/Power

ParametricPlot3D[{x, y, 
          Im[(Sqrt[(((x + I y))^2) - 1])], 
                                  {EdgeForm[],  SurfaceColor[
              Hue[0.22]]}}, 
             {x, -2, 2}, {y, -2, 
          2}, 
               PlotPoints  ->  100,  BoxRatios -> {1, 1, 0.48}];

Ability and wishes to make such site

For the first time in the history of mathematical functions due to hardware and software advances there is the possibility to create a complete listing of all identities, equations, theorems, and inequalities about mathematical functions.

The structure of this site

General statements

The site was designed to be similar in nature to existing handbooks that detail special mathematical functions, Currently it includes near 270 mathematical functions divided on the following main groups:

ElementaryFunctions,                           9231  formulas

Constants                                                  586

Bessel, Airy, Struve Functions            2186

Integer Functions                                  1334

Polynomials                                            1657

Gamma, Beta, Erf                                  3822

Hypergeometric Functions               11098

Elliptic Integrals                                       687

Elliptic Functions                                   4045

Zeta Functions and Polylogarithms     994

Mathieu Functions                                   139

Complex Components                            447

Number Theory Functions                      782

37008 formulas

Function categories

Elementary Functions

Power Functions

Sqrt

Power

Exponential Function

Exp

Logarithmic Functions

Log

Log

Trigonometric Functions

Sin

Cos

Tan

Cot

Csc

Sec

Inverse Trigonometric Functions

ArcSin

ArcCos

ArcTan

ArcTan

ArcCot

ArcCsc

ArcSec

Hyperbolic Functions

Sinh

Cosh

Tanh

Coth

Csch

Sech

Inverse Hyperbolic Functions

ArcSinh

ArcCosh

ArcTanh

ArcCoth

ArcCsch

ArcSech

Product Logarithms

ProductLog

ProductLog

Roots

Root

Max and Min

Max

Min

Constants

Imaginary Unit

I

Classical Constants

GoldenRatio

Pi

Degree

E

EulerGamma

Catalan

Glaisher

Khinchin

Symbols

Indeterminate

Infinity

ComplexInfinity

DirectedInfinity

Bessel, Airy, Struve Functions

Bessel Functions

BesselJ

BesselI

BesselY

BesselK

Airy Functions

AiryAi

AiryBi

AiryAiPrime

AiryBiPrime

AiryBiPrime.nb

Struve Functions

StruveH

StruveL

Integer Functions

Rounding and Congruence Functions

Floor

Ceiling

Round

IntegerPart

FractionalPart

Mod

Quotient

GCD and LCM

GCD

ExtendedGCD

LCM

Fibonacci Numbers

Fibonacci

Euler and Bernoulli Numbers

EulerE

BernoulliB

Stirling Numbers

StirlingS1

StirlingS2

Partitions

PartitionsP

PartitionsQ

Tensor Functions

DiscreteDelta

DiscreteDelta

KroneckerDelta

Signature

Polynomials

Classical Orthogonal Polynomials

HermiteH

LaguerreL

LegendreP

ChebyshevT

ChebyshevU

JacobiP

Associated Polynomials

LegendreP

LaguerreL

GegenbauerC

SphericalHarmonicY

Other Polynomials

Cyclotomic

Fibonacci

EulerE

BernoulliB

Gamma, Beta, Erf

Factorials

Factorial

Factorial2

Binomials

Binomial

Multinomial

Gamma Functions

Gamma

Gamma

Gamma

GammaRegularized

GammaRegularized

Pochhammer

LogGamma

InverseGammaRegularized

InverseGammaRegularized

Differentiated Gamma Functions

PolyGamma

PolyGamma

HarmonicNumber

HarmonicNumber

Beta Functions and Inverses

Beta

Beta

Beta

BetaRegularized

BetaRegularized

InverseBetaRegularized

InverseBetaRegularized

Probability Integrals and Inverses

Erf

Erf

Erfc

Erfi

InverseErf

InverseErf

InverseErfc

Fresnel Integrals

FresnelS

FresnelC

Exponential Integrals

ExpIntegralE

ExpIntegralEi

LogIntegral

SinIntegral

CosIntegral

SinhIntegral

CoshIntegral

Hypergeometric Functions

Hermite and Laguerre Functions

HermiteH

LaguerreL

LaguerreL

Chebyshev Functions

ChebyshevT

ChebyshevU

Fibonacci

Legendre Functions

LegendreP

LegendreP

LegendreP

LegendreQ

LegendreQ

LegendreQ

Gegenbauer and Jacobi Functions

GegenbauerC

GegenbauerC

JacobiP

Hypergeometric Functions of One Variable

HypergeometricPFQ

Hypergeometric0F1

Hypergeometric0F1Regularized

HypergeometricPFQ

Hypergeometric1F1

Hypergeometric1F1Regularized

HypergeometricPFQ

Hypergeometric2F1

Hypergeometric2F1Regularized

HypergeometricPFQ

HypergeometricPFQ

HypergeometricPFQ

HypergeometricPFQ

HypergeometricPFQ

HypergeometricPFQ

HypergeometricPFQ

HypergeometricPFQRegularized

HypergeometricU

Meijer G Functions

MeijerG

MeijerG

Hypergeometric Functions of Several Variables

AppellF1

Angular Momentum Functions

SphericalHarmonicY

ClebschGordan

ThreeJSymbol

SixJSymbol

Elliptic Integrals

Complete Elliptic Integrals

EllipticE

EllipticK

EllipticPi

Incomplete Elliptic Integrals

EllipticE

EllipticF

EllipticPi

JacobiZeta

Elliptic Functions

Jacobi Theta Functions

EllipticTheta

EllipticTheta

EllipticTheta

EllipticTheta

EllipticThetaPrime

EllipticThetaPrime

EllipticThetaPrime

EllipticThetaPrime

Neville Theta Functions

NevilleThetaC

NevilleThetaD

NevilleThetaN

NevilleThetaS

Weierstrass Functions and Inverses

WeierstrassP

WeierstrassPPrime

WeierstrassSigma

WeierstrassSigma

WeierstrassZeta

WeierstrassHalfPeriods

WeierstrassInvariants

WeierstrassPHalfPeriodValues

WeierstrassZetaHalfPeriodValues

InverseWeierstrassP

InverseWeierstrassP

Jacobi Functions

JacobiAmplitude

JacobiCD

JacobiCN

JacobiCS

JacobiDC

JacobiDN

JacobiDS

JacobiNC

JacobiND

JacobiNS

JacobiSC

JacobiSD

JacobiSN

Inverse Jacobi Functions

InverseJacobiCD

InverseJacobiCN

InverseJacobiCS

InverseJacobiDC

InverseJacobiDN

InverseJacobiDS

InverseJacobiNC

InverseJacobiND

InverseJacobiNS

InverseJacobiSC

InverseJacobiSD

InverseJacobiSN

Modular Functions and Inverses

DedekindEta

KleinInvariantJ

ModularLambda

InverseEllipticNomeQ

EllipticNomeQ

Arithmetic Geometric Mean

ArithmeticGeometricMean

Elliptic Exp and Elliptic Log

EllipticExp

EllipticExpPrime

EllipticLog

Zeta Functions and Polylogarithms

Zeta Functions

Zeta

Zeta

RiemannSiegelTheta

RiemannSiegelZ

StieltjesGamma

LerchPhi

Polylogarithms

PolyLog

PolyLog

PolyLog

Mathieu Functions

Mathieu Functions

MathieuC

MathieuS

MathieuCPrime

MathieuSPrime

Characteristics

MathieuCharacteristicA

MathieuCharacteristicB

MathieuCharacteristicExponent

Complex Components

Abs

Arg

Re

Im

Conjugate

Sign

Number Theory Functions

Prime Factorization Related Functions

FactorInteger

Divisors

Prime

PrimePi

DivisorSigma

EulerPhi

MoebiusMu

Other Functions

JacobiSymbol

CarmichaelLambda

DigitCount

Generalized Functions

Discontinuous Functions

UnitStep

UnitStep

Functions with Singular Support

DiracDelta

DiracDelta

Sections names in notebooks (all possibilities)

Notations

Primary Definition

Specific Values

General Characteristics

Analytic Continuations

Series Representations

Integral Representations

Product Representations

Limit Representations

Continued Fraction Representations

Generating Functions

Group Representations

Differential Equations

Difference Equations

Equations Involving This Function

Transformations

Identities

Identities Involving the Group of Functions

Complex Characteristics

Differentiation

Integration

Integral Transforms

Summation

Products

Operations

Representations Through More General Functions

Representations Valid Under Restrictions

Inequalities

Zeros

Theorems

Other Identities

Applications

History

References

Unique logical precise hierarchical structure

All formulas are placed in corresponding

Section/Subsection/Subsubsection/Subsection3/Subsection4/Subsection5 position. For example, the formula

Example1:

\(∫\(z\^\(α - 1\)\) 
          CosIntegral[
            a\ z] dz == \(1\/\(2\ α\)\) \((z\^α\ \((2\ CosIntegral[
                    a\ z] + \((\(-i\)\ a\ z)\)\^\(-α\)\ Gamma[α, \(-i\)\ a\ z] + \((i\ a\ z)\)\^\(-α\)\ Gamma[α, i\ a\ z])\))\)\)

is placed in section

Integration (Section)

—Indefinite integration (Subsection)

——Involving one direct function and elementary functions  (Subsubsection)

———Involving power function  (Subsection3)

————Involving Power (Subsection4)

—————Linear argument  (Subsection5)

This is the first formula in this Subsection5. So the position of this formula is the following

Integration/  Indefinite integration/  Involving one direct function and elementary functions/  Involving power function/  Involving Power/  Linear argument/

GammaBetaErf/CosIntegral/LongTitleForSection/SerialNumber

Example4:

Each formula on this web site is given a unique numeric identifier. As this site expands and more information is added, the position of a given formula within its group may change. However, the identifier itself will remain the same, and we recommend that you use the identifier for citation purposes.

AiryBi:

http://functions.wolfram.com/03.06.06.0013.01

(03 number of category Bessel, Airy, Struve Functions)

(06 number of the function AiryBi)

(06 number of the section Series Representations)

(0013 number of formula in the section Series Representations)

(01 number of modification)

Searchers

Keyword Search

Example: "contiguous relations"

Formulas Search

Example: _?(MemberQ[#, _Cos, {0, ∞}]&) == _?(MemberQ[#, _Sin, {0, ∞}]&)

Top ten most wanted

Below we present 10 most important questions for solving which we need in help of mathematical community:

10) Historical information about mathematical functions

We are looking for more complete historical information about mathematical functions. For each, who introduced  or its notation, and when? Who made essential progress in its development---by finding basic formulas, extending its domain of definition, defining key conventions, and so on.... A particular example: who came up with the notation  for the double factorial, and when?

9) More complete tables of particular cases of generalized hypergeometric functions

We are creating the most complete tables possible of particular cases of the generalized hypergeometric functions  Any additional examples would be greatly appreciated.

8) Differential equations for the theta functions

We are looking for ordinary, nonlinear differential equations for the elliptic theta functions with respect to  for fixed .

7) Generalizations of Euler and Bernoulli polynomials, Stirling numbers, and partitions numbers for noninteger argument

How is it possible to define the values of Euler and Bernoulli polynomials  and, Stirling numbers  and,  and partitions  and  for ?

6) Continuation of derivatives of the psi-function  by order

How is it possible to extend the definition of the derivatives of the psi-function from integer parameters  to arbitrary complex values ? Such an extension might, for instance, be provided by the formula

5) Values of derivatives of the zeta function

We are looking for closed-form expressions for the derivatives of the zeta function  of arbitrary order  for integer values of argument :   .

4) Formulas for Mathieu functions and their characteristics  and

We are looking for closed formulas for the Mathieu functions and their characteristics  and

3) The inverse function for ,,,,
The equation  has the solution . Is there a good way to express the solution of the equation?

We are looking for closed formulas for the Mathieu functions and their characteristics  and

2) Series for the inverse error function

We are searching for general formulas for the series expansion of the inverse error function around 0.

1) Series for the gamma function

We are searching for general formulas for the series expansion of the gamma function near its regular points and poles . These formulas should include the psi-functions and their derivatives. 
    Question 1: How do we express coefficients  through  and  in closed form?
    Question 2: How do we express coefficients  through  and  in closed form?  
                          It may be helpful to take into account that

Notations

Open Notations notebook

General identities

Open General Identities notebook

The vision of this site

General goals

Many additional features are planned, and a considerable amount of
new information will be incorporated in the near future.

* Designing a content-based search that would enable queries for identities that include a specific function

* Creating a Mathematica package that allows the conversion of various
 identities into interactively applicable rules inside a Mathematica session

* Adding tables of data for time-consuming calculations, such as the zeros of   Zeta and PrimePi for large arguments

* Enabling interactive numeric evaluation and plotting using webMathematica technology

* Creating Mathematica programs for generating an arbitrary number of terms  for a series expansion where no closed forms are known, such as the zeros of Airy functions and the Taylor series for Mathieu functions

* Adding special functions currently not implemented in Mathematica, such as  various totient functions, Dedekind sums, Kelvin functions, Fox's H-function, and hypergeometric and Meijer G functions of several variables

* Adding detailed reference and historical information

* Listing and expalining important applications of the various functions

* Adding special tutorials, overviews, summaries, and state-of-the-art reports about various function groups and individual special functions

The site will grow and develop for a long time

We invite specialist to take part in the developing of this site. They may submit new formulas and information. After verification they can be placed in corresponding page of the site.

We have special section "Top ten most wanted" where we give the list of the 10 most important for us mathematical problems. Welcome to solve them and help us.

The graphics galleries

We plan to create and develop the graphics galleries for all mathematical functions, presented on the site.
A preview:

Example: Graphics Gallery for ArcSin

Example: Various graphics of elementary and special functions

Example: Various Riemann surfaces

Further in the future …

Example: Riemann animations

The largest tables of integrals, sums and products and etc.

We plan to create and develop the largest tables of integrals, sums and products and other such type tables

Indefinite Integrals

Currently we have prototipe of such table including near 10000 indefinite integrals. It includes all indefinite integrals from Gradshteyn's handbook, our "Integrals and Series" vols 1-3, Timofeev's, Gröbner-Hofreiter and other handbooks. We work with this collection and after several weeks we will have well done and well organized sets of integrals in web site.

Use Mathematica to find new integrals, such as

∫f(
          g(h(x))) dx

where f, g, 
      h ∈ {sin, cos, …, arcsin, arccos, …, 
          log,  …}.

Definite Integrals

Gradshteyn's handbook has near 7000 definite integrals. We think that our collection will include near 15000.

Sums

Hansen

Examples of new formulas

General statements

★ Hundreds of new formulas
★ Hundreds of old “forgotten” formulas
★ Some new formulas even for elementary functions too, such as the correct form of sin(
          z)⟺(Sqrt[1 - ((cos(z))^2)]):

★ Many unusual and interesting formulas (for the nth prime, differential equations for modular functions, elliptic integrals, ….)

Elementary functions

Sqrt

\(TraditionalForm\`\@z\_1\ \@z\_2 == \@\(z\_1\ z\_2\)\ \(exp(\(-π\)\ i\ \ ⌊\(π - Arg(z\_1) - Arg(z\_2)\)\/\(2\ π\)⌋)\)\)

(Sqrt[1 - (z^2)]) == (Sqrt[1 - z]) (Sqrt[1 + z])

(Sqrt[(z^2) - 1]) == (Sqrt[z + 1]) (Sqrt[z + 1]) /;  
      Re (z) > 0

Many people use the power-expansion formula (Sqrt[(z^2)]) == (Sqrt[z]) (Sqrt[z]) == z. But this is correct only for real z > 0 or for complex z with -(π/2) < 
          Arg(z) ≤ (π/2). Instead of this formula we suggest the completely correct formulas

\(TraditionalForm\`\@z\^2 == \@\(\(-i\)\ z\)\ \@\(i\ z\)\)

(Sqrt[-(z^2)]) == (Sqrt[ z]) (Sqrt[- z])

Also we have the corresponding restricted formula

\(TraditionalForm\`\@\(z\_1\ z\_2\) == \@z\_1\ \@z\_2 /; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ z\_1 + z\_2 ≥ 0\)

In particular, we have

(Sqrt[1 - (z^2)]) == (Sqrt[1 - z]) (Sqrt[1 + z])

however

\(TraditionalForm\`\@\(z\^2 - 1\) == \@\(z - 1\)\ \@\(z + 1\) /; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Re(z) > 
        0\)

This restriction can be removed, involving more complicated formulas which include Floor:

\(TraditionalForm\`\@\(z\_1\ z\_2\) == \@z\_1\ \@z\_2\ \(exp(π\ i\ ⌊\(π - Arg(z\_1) - Arg(z\_2)\)\/\(2\ π\)⌋)\)\)

Sin

\(TraditionalForm\`sin(z) == C \@\( 1 - \(cos\^2\)(z)\) /; 
        C = \(\(\((\(-1\))\)\^⌊\(Re(z)\)\/π⌋\)\(\ \)\((1 - \((1 + \((\(-1\))\)\^\(⌊\(Re(z)\)\/π⌋ + ⌊\(-\(\(Re(z)\)\/π\)\)⌋\))\)\ \(θ(\(-\(Im(
                    z)\)\))\))\)\(\ \)\)\)

ArcSin

The following well-known formula

(sin^(-1))(sin(z)) == 
        z /; -(π/2) < 
          Re(z) < (π/2) ∨ (Re(z) == -(π/2) ∧ 
            Im(z) ≥ 
              0) ∨ (Re(z) == (π/2) ∧ 
            Im(z) ≤ 0)

is generalized for arbitrary values of argument:

\(TraditionalForm\`\(sin\^\(-1\)\)(
        sin(z)) == \((\(-1\))\)\^⌊\(Re(z)\)\/π - 1\/2⌋\ \((\((1 + \((\(-1\))\)\^\(⌊\(Re(z)\)\/π + 1\/2⌋ - ⌈\(Re(z)\)\/π + 1\/2⌉\))\)\ \(θ(\(-\(Im(z)\)\))\) - 
            1)\)\ \((z - π\ ⌊\(Re(z)\)\/π - 
                  1\/2⌋ + π\/2\ \((1 + \((\(-1\))\)\^\(⌊\(Re(z)\)\/π + 1\/2⌋ - ⌈\(Re(z)\)\/π + 1\/2⌉\))\)\ \(θ(\(-\(Im(
                    z)\)\))\) - π)\)\)

Special functions

AiryBi

Stokes phenomena

Here is a single formula which correctly describes the asymptotic behavior of AiryBi as \(TraditionalForm\`|z| -> ∞\) in any direction.

Bi(z)∝(((e^(((2 (z^(3/2)))/3))) (Subscript[, 2])(Subscript[F, 0])((1/6),(5/6);;(3/(4 (z^(3/2))))))/((Sqrt[π]) (z^(1/4))))/;\(|Arg(
                
                z)| < π\/3 ∧ \((|z| -> ∞)\)\)

Bi(-z)∝(1/((Sqrt[π]) (z^(1/4))))(cos(((2 (z^(3/2)))/3) + (π/4)) (Subscript[, 4])(Subscript[F, 1])((1/12),(5/12),(7/12),(11/12);(1/2);-(9/(4 (z^3))))+((5 )/(48 (z^(3/2))))(Subscript[, 4])(Subscript[F, 1])((7/12),(11/12),(13/12),(17/12);(3/2);-(9/(4 (z^3)))) sin(((2 (z^(3/2)))/3) + (π/4)))/;\(|Arg(
                z)| < \(2\ π\)\/3 ∧ \((|z| -> ∞)\)\)

Generally this behavior is described using different formulas for different zones, but with Mathematica we found this single unified formula.

Bi(z)∝(1/(2 (Sqrt[π])))(((-(z^3)))^(-5/12)) ((((-1))^(5/12)) \(e\^\(\(-\(\(\(\ \)\(2\ i\)\)\/3\)\)\ \@\(-z\^3\)\)\) \((1\/\@\(-1\)\%3\ \@\(-z\^3\)\%3 + z)\) (Subscript[, 2])(Subscript[F, 0])((1/6),(5/6);;\(\(3\ i\)\/\(4\ \@\(-z\^3\)\)\))-(((-1))^(7/12)) \(e\^\(\(2  i\)\/3\ \ \@\(-z\^3\)\)\) (((-1)^(1/3)) ((-(z^3))^(1/3)) + z) (Subscript[, 2])(Subscript[F, 0])((1/6),(5/6);;\(-\(\(3\ i\)\/\(4\ \@\(-z\^3\)\)\)\)))/;\((|z| -> ∞)\)

WeierstrassP (loooong formula)

Here is a new equation describing the double argument formula for Weierstrass (℘^′) functions.

Addition Formula

℘'(u + v, (Subscript[g, 2]), (Subscript[g, 3]))==\(4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ g\_2\%9 + 4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%9 + 
          4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%9 - 12\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%9 + 12\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%9 + 
          12\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%9 - 
          3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ g\_2\%9 - 
          3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%9 - 
          3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%9 + 
          66\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%9 + 
          26\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ g\_2\%9 - 
          26\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%9 - 
          26\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%9 + 
          66\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%9 + 
          66\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%9 - 
          3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ g\_2\%9 - 
          3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%9 - 
          3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%9 - 
          66\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%9 + 
          198\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%9 - 
          66\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%9 - 
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          12\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%9 + 
          12\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%9 - 
          66\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%9 + 
          66\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%9 + 
          66\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%9 - 
          
          66\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\ \(\(℘\^′\)(
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          12\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ g\_2\%6 - 
          12\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%6 - 
          12\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%6 - 
          36\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%6 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%6 - 
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          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          114\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          114\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          96\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          324\ g\_3\%2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          288\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 - 
          96\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 + 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%6 + 
          72\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%6 + 
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              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%6 - 72\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          72\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
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              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          540\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          432\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          540\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          972\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
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              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          972\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%6 - 
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          114\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          114\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 + 
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              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          648\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          3132\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          648\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 + 
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          216\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          486\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          3132\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          486\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 + 
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              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 + 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 - 
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              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%6 + 
          12\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ g\_2\%6 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
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          12\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          12\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 + 
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              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          540\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          432\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          1584\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 + 
          648\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          1584\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 + 
          648\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          1512\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          540\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          1512\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 + 
          96\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          486\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          36\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          288\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          540\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          486\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          3132\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          486\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          540\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          1512\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          288\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 - 
          11664\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%6 + 
          972\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\ g\_2\%6 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 + 
          96\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          972\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          972\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 + 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 + 
          96\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%6 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, 
              g\_3)\)\ g\_2\%6 + 
          3888\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, 
              g\_3)\)\ g\_2\%6 - 
          15\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          15\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          15\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          90\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          225\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          300\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          225\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          90\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8\ g\_2\%3 - 
          102\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          648\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          102\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          102\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          810\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          1686\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          864\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          648\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          1686\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          648\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          810\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 - 
          6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          864\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7\ g\_2\%3 + 
          8748\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          15\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          1296\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          8748\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          7614\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          15\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          15\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          1296\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          1650\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          1296\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          4386\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          8748\ g\_3\%4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          7614\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          3282\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          
          7614\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          4386\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          2592\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          3564\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          1650\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          30132\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 + 
          6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          3564\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6\ g\_2\%3 - 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          6804\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ g\_2\%3 + 90\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          810\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          4386\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          6804\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          6462\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          6804\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          2592\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          6462\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          27216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          18468\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          4386\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          27216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          
          23328\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          34992\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          810\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 - 
          2592\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          34992\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          90\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          26244\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          18468\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5\ g\_2\%3 + 
          7614\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          34992\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          32805\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          225\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          1686\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          864\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          7614\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          3282\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          7614\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          6462\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          27216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          18468\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          34992\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          3282\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          106272\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          34992\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          151632\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          1686\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          27216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          70956\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          32805\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          225\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          32805\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          151632\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          128304\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          864\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          18468\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 + 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4\ g\_2\%3 - 
          648\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ g\_2\%3 - 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ g\_2\%3 - 
          21870\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          300\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          648\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 - 
          1686\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          648\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          4386\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          2592\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          3564\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          
          4386\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          27216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          23328\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          34992\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 - 
          1686\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          27216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          70956\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          21870\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          300\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          2592\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          21870\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          70956\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          81648\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          34992\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          81648\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          3564\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3\ g\_2\%3 + 
          
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          32805\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          225\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          810\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          1650\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          30132\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          810\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          2592\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          34992\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          32805\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          225\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          32805\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          151632\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          128304\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          432\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          
          34992\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          81648\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          30132\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          128304\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 + 
          196830\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2\ g\_2\%3 - 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\ g\_2\%3 + 
          90\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 - 
          6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          864\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 - 
          6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          3564\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          90\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          
          26244\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          18468\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          864\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          18468\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 - 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          3564\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\ g\_2\%3 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, 
              g\_3)\)\ g\_2\%3 - 
          65610\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, 
              g\_3)\)\ g\_2\%3 + 
          4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9 - 
          4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          36\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          144\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          336\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          336\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          144\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^9 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8 - 
          324\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8 + 
          36\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          216\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          216\ g\_3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          936\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          1080\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          324\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          324\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          3960\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          4860\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          936\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          6480\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          4860\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 + 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          1080\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 - 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^8 + 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7 + 
          8748\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7 - 
          144\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 + 
          936\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 + 
          1080\ g\_3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          2052\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          4212\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          7992\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          11844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          20088\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          11844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          
          11664\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          14580\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          2052\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          52164\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          8748\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 + 
          936\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          20088\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          8748\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          52164\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          144\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          7992\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          14580\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 + 
          1080\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^7 - 
          78732\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6 - 324\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6 - 
          
          78732\ g\_3\%6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6 - 34992\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6 - 
          122472\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6 + 
          336\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          324\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          2844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          324\ g\_3\%2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          11844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          20088\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          78732\ g\_3\%6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          34992\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          48684\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          162648\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          34992\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          142236\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          11844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          102600\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          175284\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          58320\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          122472\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          2844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          162648\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          122472\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          536868\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          501552\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          336\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          20088\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          175284\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          384912\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          166212\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          142236\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          
          157464\ g\_3\%5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          501552\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 - 
          603612\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          58320\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          166212\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^6 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5 + 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5 + 
          314928\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5 + 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5 - 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 + 
          
          3960\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 + 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          11844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          14580\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          11844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          102600\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          175284\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          58320\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 + 
          3960\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          102600\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          131220\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          203148\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          244944\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          203148\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          419904\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          314928\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 + 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          175284\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          314928\ g\_3\%5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          419904\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          551124\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          14580\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          244944\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          551124\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, 
              
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          236196\ g\_3\%6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          58320\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^5 - 
          122472\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4 - 
          472392\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4 - 
          472392\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4 + 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          2844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          4860\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          2052\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          52164\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          122472\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          2844\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          162648\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          122472\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          536868\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          501552\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          504\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          12744\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          203148\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          419904\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          472392\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          536868\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          472392\ g\_3\%5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          1924560\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          1810836\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          52164\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          419904\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          761076\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          472392\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          4860\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          472392\ g\_3\%6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          501552\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          1810836\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          1732104\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^4 + 
          8748\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3 + 
          314928\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3 + 
          551124\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3 - 
          336\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 + 
          936\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 + 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 + 
          6480\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          8748\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 + 
          936\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          20088\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          8748\ g\_3\%4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          52164\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          336\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          20088\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          175284\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          384912\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          166212\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          314928\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 + 
          2376\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          175284\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          
          314928\ g\_3\%5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          419904\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          551124\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          52164\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          419904\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          761076\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          551124\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 + 
          6480\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          551124\ g\_3\%6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          384912\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          761076\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          1102248\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          551124\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          1102248\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          166212\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^3 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2 - 
          472392\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2 + 
          144\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          4860\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          144\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          7992\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          14580\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          2160\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          142236\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          157464\ g\_3\%5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          501552\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          603612\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          14580\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          244944\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          551124\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          472392\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          4860\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          472392\ g\_3\%6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          501552\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          1810836\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          1732104\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          11664\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          551124\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          1102248\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          603612\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          1732104\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 - 
          708588\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\)\^2 + 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\) - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          36\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^9\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) + 
          1080\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) + 
          1080\ g\_3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          
          4212\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          58320\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          166212\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) + 
          1944\ g\_3\%2\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^8\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          236196\ g\_3\%6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          58320\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          23328\ g\_3\%3\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          131220\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) + 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          166212\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          26244\ g\_3\%4\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^7\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^5\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^3\ \(\(℘\^′\)(v; g\_2, g\_3)\)\^2\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) - 
          157464\ g\_3\%5\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^6\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, g\_3)\) + 
          236196\ g\_3\%6\ \(\(℘\^′\)(u; g\_2, g\_3)\)\^4\ \(\(℘\^′\)(v; g\_2, 
              g\_3)\)\ \(\(℘\^′\)(u + v; g\_2, 
              g\_3)\)\)

Style for body text

HypergeometricPFQ (3F2)

(Subscript[, 3])(Subscript[F, 2])((Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3]);b,(Subscript[b, 2]);z)==(((Subscript[B, 1]) + (Subscript[C, 1]) z)/(z - 1))(Subscript[, 3])(Subscript[F, 2])((Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3]);b + 1,(Subscript[b, 2]);z)+(((Subscript[B, 2]) + (Subscript[C, 2]) z)/(z - 1)) (Subscript[, 3])(Subscript[F, 2])((Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3]);b + 2,(Subscript[b, 2]);z)+(((Subscript[C, 3]) z)/(z - 1)) (Subscript[, 3])(Subscript[F, 2])((Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3]);b + 3,(Subscript[b, 2]);z)/;\(B\_1 == \(b\_2 - 2  b - 2\)\/b ∧ 
          C\_1 == \(3\ b - a\_1 - a\_2 - a\_3 + 3\)\/b ∧ 
          B\_2 == \(b - b\_2 + 2\)\/b ∧ 
          C\_2 == \(\((2\ b + 3)\)\ \((a\_1 + a\_2 + a\_3)\) - 3\ b\^2 - 9\ b - a\_1\ a\_2 - a\_1\ a\_3 - a\_2\ a\_3 - 7\)\/\(b\ \((b + 1)\)\) ∧ C\_3 == \(\((b - a\_1 + 2)\)\ \((b - a\_2 + 2)\)\ \((b - a\_3 + 2)\)\)\/\(b\ \((b + 1)\)\ \((b + 2)\)\)\)

How to derive new formulas and make new graphics?

How to find new formulas?

Mathematica's strength comes from powerful symbolic algebraic algorithms and ability to handle large amounts of data.
The "direct" special function knowledge is largely of a look-type and often not directly useful for deriving new formulas.
The most powerful symbolic algebraic algorithms are:
* Gröbner basis (Solve)
* (modular) null space computations
Combined with verified high-precision arithmetic these are very powerful tools.

Example 1: Contiguous relations for \(TraditionalForm\`\(\_4\) F\_3\)

allTriples = 
With[{hypergeos = 
{(* all possible hypergeometric functions with parameters ±1 *)
HypergeometricPFQ[{a, b, c, d}, {e, f, g}, z],
HypergeometricPFQ[{a - 1, b, c, d}, {e, f, g}, z],
HypergeometricPFQ[{a + 1, b, c, d}, {e, f, g}, z],
HypergeometricPFQ[{a, b - 1, c, d}, {e, f, g}, z],
HypergeometricPFQ[{a, b + 1, c, d}, {e, f, g}, z],
HypergeometricPFQ[{a, b, c - 1, d}, {e, f, g}, z],
HypergeometricPFQ[{a, b, c + 1, d}, {e, f, g}, z],
HypergeometricPFQ[{a, b, c, d - 1}, {e, f, g}, z],
HypergeometricPFQ[{a, b, c, d + 1}, {e, f, g}, z],
HypergeometricPFQ[{a, b, c, d}, {e - 1, f, g}, z],
HypergeometricPFQ[{a, b, c, d}, {e + 1, f, g}, z],
HypergeometricPFQ[{a, b, c, d}, {e, f - 1, g}, z],
HypergeometricPFQ[{a, b, c, d}, {e, f + 1, g}, z],
HypergeometricPFQ[{a, b, c, d}, {e, f, g - 1}, z],
HypergeometricPFQ[{a, b, c, d}, {e, f, g + 1}, z]
}}, (* the triples *)
Flatten[Table[{hypergeos[[i]], hypergeos[[j]], hypergeos[[k]]},
             {i, Length[hypergeos]}, 
             {j, i + 1, Length[hypergeos]}, 
             {k, j + 1, Length[hypergeos]}], 2]];

Length[allTriples]

makeContiguousRelation[triple_, o_]:=
Module[{eqs, vars, ser, preSol,  rhsVars, rhsRule, sol},
(* the ansatz for the contiguous relation *)
eqs = Sum[((α[k, 0, "0"] + α[k, 0, "a"] a + 
            α[k, 0, "b"] b + α[k, 0, "c"] c + 
            α[k, 0, "d"] d + α[k, 0, "e"] e + 
            α[k, 0, "f"] f + α[k, 0, "g"] g) +
           (α[k, 1, "0"] + α[k, 1, "a"] a + 
            α[k, 1, "b"] b + α[k, 1, "c"] c + 
            α[k, 1, "d"] d + α[k, 1, "e"] e + 
            α[k, 1, "f"] f + α[k, 1, "g"] g)
            ) z triple[[k]], {k, 3}];
(* make series expansion *)
ser = CoefficientList[Series[eqs, {z, 0, o}], z]; 
tab = Flatten @ Table[ser /. Apply[Rule, Transpose[{{a, b, c, d, e, f, g}, 
                             Table[Random[Integer, {1, 100}], {7}]}], {1}], {60}];
tab1 = Numerator[Together[#]]& /@ tab; 
αs = Cases[eqs, _α, Infinity] // Union;
preSol = Solve[# == 0& /@ tab1, αs];
If[preSol =!= {}, 
(* the undetermined α[…] *)
rhsVars = Union[Flatten[Cases[#, _α, Infinity]& /@ (Last /@ preSol[[1]])]];
(* use value 1 for the undetermined α[…] *)                              
rhsRule = (# -> 1)& /@ rhsVars;
(* values for the α[…] *)  
eqs /. preSol /. rhsRule, {}]]

Off[Solve::svars];
cr1 = makeContiguousRelation[allTriples[[15]], 5]

SeedRandom[111];
cr1 /. Rule @@@ Transpose[{{a,b,c,d,e,f,g,z}, 
                          SetPrecision[Table[Random[Complex],{8}], 100]}]

Example 2: Find ODE for bivariate hypergeometric function Appel (Subscript[F, 1])

eqs0 = 
      With[{w = F1[z1, z2]}, {z1 (1 - z1) (∂_(z1, z1))w + 
            z2 (1 - 
                  z1) (∂_(z1, z2))w + (c - (1 + a + 
                        b1) z1) (∂_z1) w - 
            b1 z2 (∂_z2) w - a b1 w, 
          z2 (1 - z2) (∂_(z2, z2))w + 
            z1 (1 - 
                  z2) (∂_(z1, z2))w + (c - (1 + a + 
                        b2) z2) (∂_z2) w - 
            b2 z1 (∂_z1) w - a b2 w}]

eqs1 = {eqs0[[1]], (∂_z1) eqs0[[1]], 
          eqs0[[2]], (∂_z1) 
              eqs0[[2]], (∂_z2) 
              eqs0[[1]], (∂_(z1, z1))eqs0[[2]], (∂_(z1, z2))eqs0[[1]]};

Union[Cases[eqs1,Derivative[_, _?(#1 =!= 0 &)][F1][_, _],∞]]

eliminate[eqsOld_, eqNumber_, var_] := 
      Factor[Numerator[
            Together[
              Delete[eqsOld, 
                  eqNumber] /. Solve[
                    eqsOld[[eqNumber]] == 
                      0, var][[1]]]]] /. _Plus?(FreeQ[#1, F1, {0, ∞}, 
                  Heads -> True] &) -> 1

eqsT = eqs1;
While[(derivativesToEliminate=Union[Cases[eqsT,Derivative[_, _?(#1 =!= 0 &)][F1][_, _],∞]])=!={},nextDerivative = 
          derivativesToEliminate[[-1]]; 
        eqsT = eliminate[
            eqsT, Union[
                First /@ Position[eqsT, 
                    nextDerivative]][[-1]], nextDerivative]]

TraditionalForm[\((ode = 
              Collect[eqsT, Table[∂\_{z1, k}w[z1, z2], {k, 0, 3}], 
                FullSimplify])\)/.{Derivative[d1_, 0][w][z1, z2]:>Derivative[d1][W][z1],w[z1, z2] -> W[z1]}/.{b1 -> (Subscript[b, 1]), b2 -> (Subscript[b, 2]), 
          z1 -> (Subscript[z, 1]), z2 -> (Subscript[z, 2]), F1 -> (Subscript[F, 1])}]

SeedRandom[111];
ode/.{Derivative[d1_, d2_][F1][z1, z2]:>(∂_({z1, d1}, {z2, d2}))AppellF1[
                  a, b1, b2, c, z1, z2],F1[z1, z2] :> 
              AppellF1[a, b1, b2, c, z1, z2]}/.Apply[Rule, 
        Transpose[{{a, b1, b2, c, z1, z2}, 
            Table[Random[Real, {-1/2, 1/2}, 
                50], {6}]}], {1}]

How to visualize Riemann surfaces?

Example: (℘^((-1)))(z; 1 + i, 
          1 - 3  i).

ContourPlot[Im[InverseWeierstrassP[zr + I zi, 
                 {g2, g3}]],
            {zr, -2, 2}, {zi, -2, 2}, 
            PlotPoints -> 120,
            ColorFunction -> (Hue[2.8 #]&), 
            Contours -> 30];

Many complicated techniques used: Differential equations, functional equations, ...

Easy way: Display  {Re(z),  Im(z), f(w)}= {Re(z(w)),  Im(z(w)), f(w)}

f[z_] = WeierstrassP[z, {g2, g3}];
pp = 49;
points = Table[{Re[f[wr + I wi]], Im[f[wr + I wi]], 
                wr + I wi},
               {wr, -2.5, 2.5, 5/pp}, 
               {wi, -2.5, 2.5, 5/pp}];

prePolys = 
Table[Polygon[{points[[i, j]], points[[i, j + 1]], 
               points[[i + 1, j + 1]], 
               points[[i + 1, j]]}],
              {i, pp}, {j, pp}] // Flatten;

polysRe = Map[Re, prePolys, {-1}];
polysIm = Apply[{#1, #2, Im[#3]}&, prePolys, {-2}];

Problem: Polygons "through ∞"

infPolyQ[Polygon[l_], δMax_] :=
Max[#.#& /@ Apply[Subtract, Partition[Append[l, First[l]], 2, 1], {1}]] > δMax^2

makeHole[poly_, fac_] := 
Module[{mp = Plus @@ poly[[1]]/4, p1, p2, p3, p4, pn1, pn2, pn3, pn4},
{p1, p2, p3, p4} = poly[[1]]; pn1 = mp + fac(p1 - mp);
pn2 = mp + fac(p2 - mp); pn3 = mp + fac(p3 - mp);
pn4 = mp + fac(p4 - mp); {{EdgeForm[], 
{Polygon[{p1, p2, pn2, pn1}], Polygon[{p2, p3, pn3, pn2}], 
 Polygon[{p3, p4, pn4, pn3}], Polygon[{p4, p1, pn1, pn4}]}}, 
{Thickness[0.001], GrayLevel[0], Line[{pn1, pn2, pn3, pn4, pn1}]}}];

Show[GraphicsArray[
Graphics3D[{EdgeForm[{Thickness[0.001], Hue[0.23]}],
            SurfaceColor[Hue[0.02], Hue[0.12], 2.6],
            makeHole[#, 0.7]& /@ DeleteCases[#, _?(infPolyQ[#, 1]&)]},
     BoxRatios -> {1, 1, 2}, PlotRange -> All, 
     ViewPoint -> {2, -2, 1}]& /@ {polysRe, polysIm}]];

How is the website made and maintained?

* All source files are Mathematica notebooks containing all the in the webiste shown (and much more) information

* Various Mathematica programs to create
   - Mathematica notebooks to be downloaded
   - Mathematica notebooks to be printed to a file
                                                  (Underscript[⟶, Distiller]) pdf files
   - large directories containing the raw information of the web pages 
                                                   (Underscript[⟶, Perl]) html pages

* Possibility to make changes in nearly 300 files with more than 35000 formulas in a programmatic way is essential

* Possibility to check many of the more than 35000 formulas in a programmatic way is essential