E has complex multiplication, it is well known that E and Â¯Ï�E,p are mod- ular. Since E has good reduction at p, it can be shown that the discriminant of contains no nontrivial p-th roots of unity. One can show F is prime to p and that all of the hypotheses of Theorem 5.4 are satisï¬�ed with Â¯Ï� = Â¯Ï�E,p âŠ— k. By our assumptions on Eâ€², Ï�Eâ€²,p âŠ— O )-lifting of Â¯Ï�, and we conclude (using the D same reasoning as in the proofs of Propositions 3.3 and 4.3) that Ï�Eâ€²,p is modular and hence Eâ€² is modular.\n\nO\n\nis a (\n\n,\n\nO\n\nRemarks. (i) The elliptic curves Eâ€² of Corollary 5.5 are not semistable.\n\n(ii) Suppose E and p are as in Corollary 5.5 and p = 3 or 5. As in Appendix B.2 one can show that the elliptic curves Eâ€² over Q with good reduction at p and with Â¯Ï�Eâ€²,p isomorphic to Â¯Ï�E,p give inï¬�nitely many C-isomorphism classes.\n\nExample. Take E to be the elliptic curve deï¬�ned by\n\ny2 = x3\n\nx2\n\n3x\n\n1.\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nThen E has complex multiplication by Q(âˆš Deï¬�ne polynomials\n\n2), and E has good reduction at 3.\n\nâˆ’\n\n1512t3 3, a4(t) = a6(t) = 40824t6 + 31104t5 + 8370t4 + 504t3\n\n2430t4\n\n396t2\n\n56t\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\n148t2\n\n24t\n\n1,\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nQ let Et be the elliptic curve\n\nand for each t\n\nâˆˆ\n\ny2 = x3\n\nx2 + a4(t)x + a6(t)\n\nâˆ’\n\nQ, Â¯Ï�Et,3 is isomorphic to (note that E0 = E). It can be shown that for every t 0 or 1 (mod 3) (or more generally if t = 3a/b or t = 3a/b + 1 Â¯Ï�E,3. If t with a and b integers and b not divisible by 3), then Et has good reduction at 3, for instance because the discriminant of Et is\n\nâˆˆ\n\nZ and t\n\nâˆˆ\n\nâ‰¡\n\n29(27t2 + 10t + 1)3(27t2 + 18t + 1)3.\n\nThus for these values of t, Corollary 5.5 shows that Et is modular and so is any elliptic curve over Q isomorphic over C to Et, i.e., any elliptic curve over Q with j-invariant equal to\n\n3\n\n4(27t2 + 6t + 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t +
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: E has complex multiplication, it is well known that E and Â¯Ï�E,p are mod- ular. Since E has good reduction at p, it can be shown that the discriminant of contains no nontrivial p-th roots of unity. One can show F is prime to p and that all of the hypotheses of Theorem 5.4 are satisï¬�ed with Â¯Ï� = Â¯Ï�E,p âŠ— k. By our assumptions on Eâ€², Ï�Eâ€²,p âŠ— O )-lifting of Â¯Ï�, and we conclude (using the D same reasoning as in the proofs of Propositions 3.3 and 4.3) that Ï�Eâ€²,p is modular and hence Eâ€² is modular.\n\nO\n\nis a (\n\n,\n\nO\n\nRemarks. (i) The elliptic curves Eâ€² of Corollary 5.5 are not semistable.\n\n(ii) Suppose E and p are as in Corollary 5.5 and p = 3 or 5. As in Appendix B.2 one can show that the elliptic curves Eâ€² over Q with good reduction at p and with Â¯Ï�Eâ€²,p isomorphic to Â¯Ï�E,p give inï¬�nitely many C-isomorphism classes.\n\nExample. Take E to be the elliptic curve deï¬�ned by\n\ny2 = x3\n\nx2\n\n3x\n\n1.\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nThen E has complex multiplication by Q(âˆš Deï¬�ne polynomials\n\n2), and E has good reduction at 3.\n\nâˆ’\n\n1512t3 3, a4(t) = a6(t) = 40824t6 + 31104t5 + 8370t4 + 504t3\n\n2430t4\n\n396t2\n\n56t\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\n148t2\n\n24t\n\n1,\n\nâˆ’\n\nâˆ’\n\nâˆ’\n\nQ let Et be the elliptic curve\n\nand for each t\n\nâˆˆ\n\ny2 = x3\n\nx2 + a4(t)x + a6(t)\n\nâˆ’\n\nQ, Â¯Ï�Et,3 is isomorphic to (note that E0 = E). It can be shown that for every t 0 or 1 (mod 3) (or more generally if t = 3a/b or t = 3a/b + 1 Â¯Ï�E,3. If t with a and b integers and b not divisible by 3), then Et has good reduction at 3, for instance because the discriminant of Et is\n\nâˆˆ\n\nZ and t\n\nâˆˆ\n\nâ‰¡\n\n29(27t2 + 10t + 1)3(27t2 + 18t + 1)3.\n\nThus for these values of t, Corollary 5.5 shows that Et is modular and so is any elliptic curve over Q isomorphic over C to Et, i.e., any elliptic curve over Q with j-invariant equal to\n\n3\n\n4(27t2 + 6t + 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t +