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<p>We establish a classification of the values of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( N \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation> for which an elliptic curve defined over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \mathbb {Q} \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> with square discriminant admits an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( N \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation>-isogeny. Furthermore, we determine the values of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( N \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation> for which two elliptic curves defined over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( \mathbb {Q} \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>, both possessing square discriminants, are <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( N \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation>-isogenous. In both cases, we explicitly parametrize the corresponding <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( j \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>j</mi> </math></EquationSource> </InlineEquation>-invariants of the elliptic curves associated with these problems.</p>

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

  • 期刊论文

  • 发表日期：

    2025年11月19日

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  • Enrique González-Jiménez

摘要

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