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 p of K, there are p-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.
- Prime isogenous discriminant ideal twins
- Alexander J. Barrios - University of St. Thomas - MinnesotaMaila Brucal-Hallare - United States Air Force AcademyAlyson Deines - Center for Communications Research, San Diego, CA, USAPiper Harris - Lafayette CollegeManami Roy - Lafayette College
- Journal of number theory
- Elsevier Inc
- College of Arts and Sciences; Mathematics
- English
- Journal article
- 991015438146803691