## The modular curve $X_{315}$

Curve name $X_{315}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $24$ $X_{96}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{315}$ minimally covers $X_{96}$, $X_{117}$, $X_{159}$
Curves that minimally cover $X_{315}$ $X_{475}$, $X_{483}$, $X_{613}$, $X_{614}$
Curves that minimally cover $X_{315}$ and have infinitely many rational points.
Model A model was not computed. This curve is covered by $X_{159}$, 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