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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> (where <math>\tau\in\mathbb{C}</math> such that <math>\operatorname{Re}\tau=0</math> and <math>\operatorname{Im}\tau>0</math>). If <math>u=\sqrt[8]{\alpha}</math> and <math>v=\sqrt[8]{\beta}</math>, the modular equations of degrees <math>2,3,5,7</math> are, respectively,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 103–109, 134</ref>
:<math>(1+u^4)^2v^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>
:<math>(1-u^8)(1-v^8)-(1-uv)^8=0.</math>
==Elliptic modulus==
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