The modular curve $X_{619}$

Curve name $X_{619}$
Index $96$
Level $32$
Genus $3$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 25 & 18 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 25 & 25 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 25 & 11 \\ 2 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{91}$ $16$ $48$ $X_{288}$
Meaning/Special name
Chosen covering $X_{288}$
Curves that $X_{619}$ minimally covers $X_{288}$
Curves that minimally cover $X_{619}$
Curves that minimally cover $X_{619}$ and have infinitely many rational points.
Model $x^4 - x^2y^2 - 2x^2z^2 - y^3z + 2yz^3 = 0$
 Rational point Image on the $j$-line $(-2 : -4 : 1)$ $\frac{-631595585199146625}{218340105584896}$ $(0 : 1 : 0)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(1 : 1 : 0)$ $16581375 \,\,(\text{CM by }-28)$ $(2 : -4 : 1)$ $\frac{-631595585199146625}{218340105584896}$ $(-1 : 1 : 0)$ $16581375 \,\,(\text{CM by }-28)$
Comments on finding rational points This curve admits a family of etale double covers of genus 5. These double covers have 16 automorphisms defined over $K = \mathbb{Q}(\sqrt{2+\sqrt{2}})$ and as a consequence map to elliptic curves defined over $K$. These curves have rank two over $K$ and we use elliptic curve Chabauty.
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 303870x - 81409651$, with conductor $17918$
Generic density of odd order reductions $410657/1835008$