The modular curve $X_{353}$

Curve name $X_{353}$
Index $48$
Level $32$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $12$ $X_{36}$
$16$ $24$ $X_{118}$
Meaning/Special name
Chosen covering $X_{118}$
Curves that $X_{353}$ minimally covers $X_{118}$
Curves that minimally cover $X_{353}$ $X_{492}$, $X_{493}$, $X_{494}$, $X_{497}$, $X_{617}$, $X_{621}$, $X_{627}$, $X_{636}$
Curves that minimally cover $X_{353}$ and have infinitely many rational points.
Model \[y^2 = x^3 - x\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[ \infty \]
$(-1 : 0 : 1)$ \[ \infty \]
$(0 : 0 : 1)$ \[ \infty \]
$(1 : 0 : 1)$ \[ \infty \]
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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