Prime isogenous discriminant ideal twins
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913021813587968 |
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| author | Barrios, Alexander J. Brucal-Hallare, Maila Deines, Alyson Harris, Piper Roy, Manami |
| author_facet | Barrios, Alexander J. Brucal-Hallare, Maila Deines, Alyson Harris, Piper Roy, Manami |
| contents | Let $E_{1}$ and $E_{2}$ be elliptic curves defined over a number field $K$. We say that $E_{1}$ and $E_{2}$ 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 $\mathfrak{p}$ of $K$, there are $\mathfrak{p}$-minimal models for $E_{1}$ and $E_{2}$ whose discriminants are equal. This article explicitly classifies all prime-isogenous discriminant (ideal) twins over $\mathbb{Q}$. 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\in\left\{ 2,3,5,7,13\right\} $, i.e., where $X_0(p)$ 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 $\mathbb{Q}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_19183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prime isogenous discriminant ideal twins Barrios, Alexander J. Brucal-Hallare, Maila Deines, Alyson Harris, Piper Roy, Manami Number Theory 11G05, 11G07, 14K02, 14H10, 14H52 Let $E_{1}$ and $E_{2}$ be elliptic curves defined over a number field $K$. We say that $E_{1}$ and $E_{2}$ 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 $\mathfrak{p}$ of $K$, there are $\mathfrak{p}$-minimal models for $E_{1}$ and $E_{2}$ whose discriminants are equal. This article explicitly classifies all prime-isogenous discriminant (ideal) twins over $\mathbb{Q}$. 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\in\left\{ 2,3,5,7,13\right\} $, i.e., where $X_0(p)$ 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 $\mathbb{Q}$ 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. |
| title | Prime isogenous discriminant ideal twins |
| topic | Number Theory 11G05, 11G07, 14K02, 14H10, 14H52 |
| url | https://arxiv.org/abs/2402.19183 |