The modular curve $X_{678}$

Curve name	$X_{678}$
Index	$96$
Level	$32$
Genus	$5$
Does the subgroup contain $-I$?	Yes
Generating matrices	$\left[ \begin{matrix} 21 & 21 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 17 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 15 & 13 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 26 \\ 8 & 1 \end{matrix}\right]$
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 $X_{678}$ minimally covers	$X_{288}$
Curves that minimally cover $X_{678}$
Curves that minimally cover $X_{678}$ and have infinitely many rational points.
Model	$$y^2 = x^3 + x^2 - 3x + 1$$ $$w^2 = 6x^2y + 16x^2 - 8xy - 16x - y^3 - 4y^2 - 2y$$
Info about rational points	Rational point Image on the $j$-line $(1 : 0 : 1 : 0)$ $$\infty$$ $(0 : -1 : 1 : 1)$ $$16581375 \,\,(\text{CM by }-28)$$ $(0 : 1 : -1 : 1)$ $$16581375 \,\,(\text{CM by }-28)$$ $(-1 : -2 : 1 : 0)$ Singular $(0 : 0 : 0 : 1)$ Singular
Comments on finding rational points	This curve admits a family of twists of etale double covers that are also modular curves. Each of these modular curves maps to one we have already computed that has finitely many rational points.
Elliptic curve whose $2$-adic image is the subgroup	None
Generic density of odd order reductions	N/A

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