Abstract
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.