Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(K_\rho/\mathbb{Q})$. The equivariant BSD conjecture states that $$\mathrm{ord}_{s=1}L(E\otimes\rho,s) \; = \; \text{multiplicity of $\rho$ in $E(K_\rho)\otimes\mathbb{C}$}.$$

When $\rho$ is the trivial representation, the above conjecture reduces to the classical BSD conjecture. In this case, as far as I understand, the conjecture is almost completely proven for rank 0 and 1, while for higher ranks we have computational evidence.

My question/reference request is: Do we also have (computational) evidence for the above conjecture when the dimension of $\rho$ is $\ge2$ and the multiplicity is also $\ge2$? I would be happy to see any numerical examples, papers, or even code. I'm having a hard time even finding elliptic curves in which a non-trivial representation appears twice in $E(K_\rho)\otimes\mathbb{C}$. I guess they exist, but as in the classical conjecture, higher multiplicity might be extremely rare.

As a special case, I'm particularly interested in examples where $\rho$ is a 2-dimensional odd irreducible Artin representation (so it is associated to a weight 1 eigenform). I have been learning about the subject, and I see that many of the known results concern such representations, as in this case one can use the Rankin-Selberg method to study the $L$-function. So this might be a source of some very good examples.

On a side note: I would be pleased to know if someone has already implemented the objects on the right-hand side of the conjecture in SageMath (or in some other software).

Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(K_\rho/\mathbb{Q})$. The equivariant BSD conjecture states that $$\mathrm{ord}_{s=1}L(E\otimes\rho,s) \; = \; \text{multiplicity of $\rho$ in $E(K_\rho)\otimes\mathbb{C}$}.$$

When $\rho$ is the trivial representation, the above conjecture reduces to the classical BSD conjecture. In this case, as far as I understand, the conjecture is almost completely proven for rank 0 and 1, while for higher ranks we have computational evidence.

My question/reference request is: Do we also have (computational) evidence for the above conjecture when the dimension of $\rho$ is $\ge2$ and the multiplicity is also $\ge2$? I would be happy to see any numerical examples, papers, or even code. I'm having a hard time even finding elliptic curves in which a non-trivial representation appears twice in $E(K_\rho)\otimes\mathbb{C}$. I guess they exist, but as in the classical conjecture, higher multiplicity might be extremely rare.

As a special case, I'm particularly interested in examples where $\rho$ is a 2-dimensional odd irreducible Artin representation (so it is associated to a weight 1 eigenform). I have been learning about the subject, and I see that many of the known results concern such representations. In this setting, one can use the Rankin-Selberg method to study the $L$-function, potentially leading to some very interesting examples.

On a side note: I would be pleased to know if someone has already implemented the objects on the right-hand side of the conjecture in SageMath (or in some other software).