Towards a classification of p²-discriminant ideal twins over number fields

Isogenous elliptic curves have the same conductor but not necessarily the same minimal discriminant ideal. In this article, we explicitly classify all p²-isogenous elliptic curves defined over a number field with the same minimal discriminant ideal for odd prime p where X₀(p²) has genus $0$, i.e., $p = 3$ or $5$. As a consequence, we give a list of all p²-isogenous discriminant (ideal) twins over Q for such p.