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<p>In this article, we consider the elliptic curve <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(E: y^2 = (x+1)(x^2+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>:</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and prove that its quadratic twists have rank zero for infinitely many squarefree integers.</p>

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On ranks of quadratic twists of elliptic curve \(\varvec{y^2{\,=\,}(x{\,+\,}1)}\) \(\varvec{(x^2{\,+\,}1)}\)

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Abhishek Juyal,
  • Mansi Tyagi

摘要

In this article, we consider the elliptic curve \(E: y^2 = (x+1)(x^2+1)\) E : y 2 = ( x + 1 ) ( x 2 + 1 ) and prove that its quadratic twists have rank zero for infinitely many squarefree integers.