Classical modular curve: Difference between revisions

Revision as of 04:27, 24 May 2006

Revision as of 04:31, 24 May 2006

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When X<sub>0</sup>(n) has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant]]. For instance, X<sub>0</sup>(11) has j-invariant -122023936/161051 = - 2¹²11^-531³, and is isomorphic to the curve y²+y = x³-x<sup>2^{-10x-20. If we substitute this value of j for y in X<sub>0}(5), we obtain two rational roots and a factor of degree four. The two rational roots correspond to isomorphism classes of curves with rational coefficients which are 5-isogenous to the above curve, but not isomorphic, having a different function field.

Specifically, we gethave the six rational points x=-122023936/161051, y=-4096/11, x=-122023936/161051, and y=-52893159101157376/11, and x=-4096/11, y=y=-52893159101157376/11, plus the three points exchanging x and y, all on X<sub>0</sup>(5), andcorresponding theseto correspondthe tosix isogenies between these three curves. If in the curve y²+y = x³-x<sup>2^{-10x-20 isomorphic to X<sub>0}(11) we substitute

:<math>x \mapsto \frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}</math>

and