2024/02/29 by Barrios, Alexander J., Brucal-Hallare, Maila, Deines, Alyson +2 · 1 citation
#11G05 #11G07 #14H10 #14H52 #14K02 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2402.19183
Let E1 and E2 be elliptic curves defined over a number field K. We say that E1 and E2 are discriminant ideal twins if they are not K-isomorphic and have the same minimal discriminant ideal and conductor. Such curves are said to be discriminant twins if, for each prime \mathfrakp of K, there are \mathfrakp-minimal models for E1 and E2 whose discriminants are equal. This article explicitly classifies all prime-isogenous discriminant (ideal) twins over ℚ. We obtain this classification as a consequence of our main results, which constructively gives all p-isogenous discriminant ideal twins over number fields where p∈\ 2,3,5,7,13\, i.e., where X0(p) has genus 0. In particular, we find that up to twist, there are finitely many p-isogenous discriminant ideal twins if and only if K is ℚ or an imaginary quadratic field. In the latter case, we provide instructions for finding the finitely many pairs of j-invariants that result in p-isogenous discriminant ideal twins. We prove our results by considering the local data of parameterized p-isogenous elliptic curves.