Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve $$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$ More precisely, $C$ is a twist of the modular curve $X_{0}(26)$. I know that Bruin and Najman have determined all quadratic points on $X_{0}(26)$ using the finiteness of the Jacobian of $X_{0}(26)$ over $\mathbb{Q}$.

Let $J$ denote the Jacobian of $C$. Then $J$ has rank 1 over $\mathbb{Q}$. It's also interesting that $C(\mathbb{Q})=\emptyset$ (this can be seen using the TwoCoverDescent function on Magma).

I would really appreciate any pointers on how to proceed.

Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve $$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$ More precisely, $C$ is a twist of the modular curve $X_{0}(26)$. I know that Bruin and Najman (https://arxiv.org/abs/1406.0655) have determined all quadratic points on $X_{0}(26)$ using the finiteness of the Jacobian of $X_{0}(26)$ over $\mathbb{Q}$.

Let $J$ denote the Jacobian of $C$. Then $J$ has rank 1 over $\mathbb{Q}$. It's also interesting that $C(\mathbb{Q})=\emptyset$ (this can be seen using the TwoCoverDescent function on Magma).

I would really appreciate any pointers on how to proceed.