The modular curve $X_{256}$

Curve name $X_{256}$
Index $48$
Level $8$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{24}$
Meaning/Special name
Chosen covering $X_{66}$
Curves that $X_{256}$ minimally covers $X_{66}$, $X_{67}$, $X_{90}$, $X_{91}$, $X_{128}$, $X_{141}$, $X_{144}$
Curves that minimally cover $X_{256}$ $X_{600}$, $X_{601}$
Curves that minimally cover $X_{256}$ and have infinitely many rational points.
Model A model was not computed. This curve is covered by $X_{53}$, which only has finitely many rational points.
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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