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Ira Leviton (talk | contribs) m Fixed a typo found with Wikipedia:Typo_Team/moss. |
4pq1injbok (talk | contribs) →Mappings: ln "conductor" |
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and factor, we get an extraneous factor of a rational function of {{mvar|x}}, and the curve {{math|''y''<sup>2</sup> + ''y'' {{=}} ''x''<sup>3</sup> − ''x''<sup>2</sup>}}, with {{mvar|j}}-invariant {{math|−2<sup>12</sup>11<sup>−1</sup>}}. Hence both curves are modular of level {{math|11}}, having mappings from {{math|''X''<sub>0</sub>(11)}}.
By a theorem of [[Henri Carayol]], if an elliptic curve {{mvar|E}} is modular then its [[conductor of an elliptic curve|conductor]], an isogeny invariant described originally in terms of [[cohomology]], is the smallest integer {{mvar|n}} such that there exists a rational mapping {{math|''φ'' : ''X''<sub>0</sub>(''n'') → ''E''}}. Since we now know all elliptic curves over {{math|'''Q'''}} are modular, we also know that the conductor is simply the level {{mvar|n}} of its minimal modular parametrization.
== Galois theory of the modular curve ==
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