The modular curve $X_{515}$

Curve name	$X_{515}$
Index	$96$
Level	$16$
Genus	$2$
Does the subgroup contain $-I$?	Yes
Generating matrices	$\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{102}$
Meaning/Special name
Chosen covering	$X_{235}$
Curves that $X_{515}$ minimally covers	$X_{235}$
Curves that minimally cover $X_{515}$
Curves that minimally cover $X_{515}$ and have infinitely many rational points.
Model	$$y^2 = x^5 - 2x^4 - 2x^2 - x$$
Info about rational points	Rational point Image on the $j$-line $(1 : 0 : 0)$ $$\infty$$ $(0 : 0 : 1)$ $$\infty$$
Comments on finding rational points	The rank of the Jacobian is 0. We use the method of Chabauty.
Elliptic curve whose $2$-adic image is the subgroup	None
Generic density of odd order reductions	N/A

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