Equivariant Birch–Swinnerton–Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over $\mathbb{Q}$ and let $K/\mathbb{Q}$ be a finite Galois extension with Galois group G. The equivariant Birch–Swinnerton–Dyer conjecture for $h^{1}(E\times_{\mathbb{Q}}K)(1)$ viewed as a motive over $\mathbb{Q}$ with coefficients in $\mathbb{Q}[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S3-extension of $\mathbb{Q}$.