In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve $X_0(14)$, using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of $X_0(14)$ whose Tate–Shafarevich group is of odd cardinality.

中文翻译：

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X _0的二次扭曲（14）

在本文中，我们证明了模块化椭圆曲线的第0级二次扭曲的显式无穷系列的Birch和Swinnerton-Dyer猜想的2部分 $X_0（14）$，使用Waldspurger公式的显式形式。我们还给出了的第1级二次扭曲的显式无限族$X_0（14）$ 其Tate–Shafarevich组的基数为奇数。