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A classification of ECM-friendly families using modular curves

Abstract : In this work, we establish a link between the classification of ECM-friendly curves and Mazur's program B, which consists in parameterizing all the families of elliptic curves with exceptional Galois image. Building upon two recent works which treated the case of congruence subgroups of prime-power level which occur for infinitely many $j$-invariants, we prove that there are exactly 1525 families of rational elliptic curves with distinct Galois images which are cartesian products of subgroups of prime-power level. This makes a complete list of rational families of ECM-friendly elliptic curves, out of which less than 25 were known in the literature. We furthermore refine a heuristic of Montgomery to compare these families and conclude that the best 4 families which can be put in $a=-1$ twisted Edwards' form are new.
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https://hal.archives-ouvertes.fr/hal-01822144
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Submitted on : Thursday, February 14, 2019 - 5:41:59 PM
Last modification on : Saturday, December 4, 2021 - 3:43:45 AM
Long-term archiving on: : Wednesday, May 15, 2019 - 8:06:35 PM

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Razvan Barbulescu, Sudarshan Shinde. A classification of ECM-friendly families using modular curves. Mathematics of Computation, American Mathematical Society, In press. ⟨hal-01822144v2⟩

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