The modular curve $X_{288}$

Curve name	$X_{288}$
Index	$48$
Level	$16$
Genus	$1$
Does the subgroup contain $-I$?	Yes
Generating matrices	$\left[ \begin{matrix} 9 & 2 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 9 & 9 \\ 2 & 7 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{91}$
Meaning/Special name
Chosen covering	$X_{91}$
Curves that $X_{288}$ minimally covers	$X_{91}$, $X_{105}$, $X_{165}$
Curves that minimally cover $X_{288}$	$X_{587}$, $X_{615}$, $X_{618}$, $X_{619}$, $X_{678}$, $X_{690}$
Curves that minimally cover $X_{288}$ and have infinitely many rational points.
Model	$$y^2 = x^3 + x^2 - 3x + 1$$
Info about rational points	$X_{288}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 + xy = x^3 + x^2 - 1163177961056300788x - 365926510640189061755312483$, with conductor $20167985106465$
Generic density of odd order reductions	$12833/57344$

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