## The modular curve $X_{312}$

Curve name $X_{312}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 10 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 4 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 7 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{97}$
Meaning/Special name
Chosen covering $X_{97}$
Curves that $X_{312}$ minimally covers $X_{97}$, $X_{110}$, $X_{150}$
Curves that minimally cover $X_{312}$ $X_{635}$, $X_{637}$, $X_{686}$, $X_{689}$
Curves that minimally cover $X_{312}$ and have infinitely many rational points.
Model $y^2 = x^3 + x^2 - 3x + 1$
Info about rational points $X_{312}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 13804780x + 191741059200$, with conductor $173600$
Generic density of odd order reductions $42979/172032$