On the squareness of the discriminant of elliptic curves with an isogeny

TL;DR

The paper classifies when an elliptic curve $E/\mathbb{Q}$ with square discriminant ($\sqrt{\Delta(E)}\in\mathbb{Q}$) admits a rational $N$-isogeny and, similarly, when two such curves can be $N$-isogenous. The authors reduce the problem to rational points on modular curves $C_N$ and $X_N$, performing genus-based analyses to determine possible $N$ and providing explicit $j$-invariant parametrizations in the admissible cases. They show that, for a single curve, $N$ must lie in $\{2,3,4,6,7,8\}$, while for pairs of curves it must lie in $\{2,3,4,7\}$, and they give CM-specific results clarifying the discriminant-square situation in that setting. The work yields concrete arithmetic criteria and parametrizations that sharpen our understanding of how discriminant squares interact with isogeny structures on elliptic curves over $\mathbb{Q}$.

Abstract

We establish a classification of the values of \( N \) for which an elliptic curve defined over \( \mathbb{Q} \) with square discriminant admits an \( N \)-isogeny. Furthermore, we determine the values of \( N \) for which two elliptic curves defined over \( \mathbb{Q} \), both possessing square discriminants, are \( N \)-isogenous. In both cases, we explicitly parametrize the corresponding \( j \)-invariants of the elliptic curves associated with these problems.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• proof
• Theorem 5
• Lemma 6
• proof