## The modular curve $X_{355}$

Curve name $X_{355}$
Index $48$
Level $32$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 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
Chosen covering $X_{118}$
Curves that $X_{355}$ minimally covers $X_{118}$
Curves that minimally cover $X_{355}$ $X_{488}$, $X_{491}$, $X_{495}$, $X_{498}$, $X_{620}$, $X_{622}$, $X_{629}$, $X_{638}$, $X_{662}$, $X_{664}$, $X_{666}$, $X_{669}$, $X_{696}$, $X_{701}$, $X_{702}$, $X_{703}$
Curves that minimally cover $X_{355}$ and have infinitely many rational points.
Model $y^2 = x^3 - 4x$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(-2 : 0 : 1)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(2 : 0 : 1)$ $\infty$
Elliptic curve whose $2$-adic image is the subgroup None