## The modular curve $X_{354}$

Curve name $X_{354}$
Index $48$
Level $32$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$ $16$ $24$ $X_{118}$
Meaning/Special name $X_{0}(32)$
Chosen covering $X_{118}$
Curves that $X_{354}$ minimally covers $X_{118}$
Curves that minimally cover $X_{354}$ $X_{490}$, $X_{496}$, $X_{617}$, $X_{621}$, $X_{629}$, $X_{638}$, $X_{663}$, $X_{665}$, $X_{667}$, $X_{668}$, $X_{697}$, $X_{698}$, $X_{699}$, $X_{700}$
Curves that minimally cover $X_{354}$ and have infinitely many rational points.
Model $y^2 = x^3 + 4x$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(2 : 4 : 1)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(2 : -4 : 1)$ $\infty$
Elliptic curve whose $2$-adic image is the subgroup None