The modular curve $X_{654}$

Curve name $X_{654}$
Index $96$
Level $32$
Genus $3$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 20 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 28 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 20 & 13 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{23}$
$8$ $24$ $X_{69}$
$16$ $48$ $X_{324}$
Meaning/Special name
Chosen covering $X_{324}$
Curves that $X_{654}$ minimally covers $X_{324}$
Curves that minimally cover $X_{654}$
Curves that minimally cover $X_{654}$ and have infinitely many rational points.
Model \[-x^3y + 2xy^3 - z^4 = 0\]
Info about rational points
Rational pointImage on the $j$-line
$(1 : 1 : 1)$ \[-3375 \,\,(\text{CM by }-7)\]
$(0 : 1 : 0)$ \[ \infty \]
$(1 : 0 : 0)$ \[ \infty \]
$(-1 : -1 : 1)$ \[-3375 \,\,(\text{CM by }-7)\]
Comments on finding rational points This curve is isomorphic to $X_{628}$.
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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