Classical modular curve: Difference between revisions

Revision as of 19:08, 25 May 2006

Revision as of 06:57, 27 May 2006

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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).

By a theorem of [[Henri Carayol]], if an elliptic curve E is modular thanthen its conductor, an isogeny invariant described originally in terms of [[cohomology]],

is the smallest integer n such that there exists a rational mapping &phi;:X₀(n)</sub> → E. Since we now know all elliptic curves over '''Q''' are modular, we also know that the conductor is simply the level n of its minimal modular parametrization.