论文标题

八颗苯胺方程和相关椭圆曲线的家族

An octic diophantine equation and related families of elliptic curves

论文作者

Choudhry, Ajai, Zargar, Arman Shamsi

论文摘要

We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4-2x_1^4x_3^4-2x_2^4x_3^4 $。这些参数溶液产生了两个epiareal三角形的示例,它们的侧面是整数的完美正方形。此外,两种参数解决方案中的每一个都导致等级的椭圆曲线家庭〜$ 5 $超过$ \ mathbb {q}(t)$。我们详细研究了两个家庭之一,并确定了该家庭的五个自由发电机。

We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.

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