Content deleted Content added
→Mappings: improving explanation of isogenies |
→Mappings: isogeny example |
||
Line 25:
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 = - 2<sup>12</sup>11<sup>-5</sup>31<sup>3</sup>, and is isomorphic to the curve y<sup>2</sup>+y = x<sup>3</sup>-x<sup>2<sup>-10x-20. If we substitute this value of j for y in X<sub>0</sup>(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 get the six rational points x=-122023936/161051, y=-4096/11, x=-122023936/161051, and y=-52893159101157376/11, x=-4096/11, y=y=-52893159101157376/11, plus the three points exchanging x and y, all on X<sub>0</sup>(5), and these correspond to isogenies. If in the curve y<sup>2</sup>+y = x<sup>3</sup>-x<sup>2<sup>-10x-20 isomorphic to X<sub>0</sup>(11) we substitute
:<math>x \mapsto \frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}</math>
and
:<math>y \mapsto y-\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}</math>
and factor, we get an extraneous factor of a rational function of x, and the curve y^2+y=x^3-x^2, with j-invariant -4096/11. Hence both curves are modular of level 11, having mappings from X<sub>0</sup>(11).
== Galois theory of the modular curve ==
| |||