Modular lambda function: Difference between revisions

Revision as of 21:14, 3 January 2022

Revision as of 23:08, 3 January 2022

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which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>

==Modular equations==

The ''modular equation of degree'' <math>p</math> (where <math>p</math> is a prime number) is an algebraic equation in <math>\alpha = \lambda (p\tau)</math> and <math>\beta =\lambda (\tau)</math>.<ref>{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=23.20.iv|title=Weierstrass Elliptic and Modular Functions}}</ref> If <math>u=\sqrt[4]{\alpha}</math> and <math>v=\sqrt[4]{\beta}</math>, the modular equations of degrees <math>2,3,5</math> are, respectively,

:<math>(1+u^8)v^8-4u^4=0,</math>

:<math>u^4-v^4+2uv(1-u^2v^2)=0,</math>

:<math>u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0.</math>

==Elliptic modulus==