## The modular curve $X_{320}$

Curve name $X_{320}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{23}$ $8$ $24$ $X_{69}$
Meaning/Special name
Chosen covering $X_{69}$
Curves that $X_{320}$ minimally covers $X_{69}$, $X_{109}$, $X_{153}$
Curves that minimally cover $X_{320}$ $X_{575}$, $X_{581}$
Curves that minimally cover $X_{320}$ and have infinitely many rational points.
Model $y^2 = x^3 + x^2 - 13x - 21$
Info about rational points $X_{320}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 219550x + 37459245$, with conductor $4046$
Generic density of odd order reductions $12833/57344$