## The modular curve $X_{328}$

Curve name $X_{328}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 9 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{71}$
Meaning/Special name
Chosen covering $X_{71}$
Curves that $X_{328}$ minimally covers $X_{71}$, $X_{113}$, $X_{155}$
Curves that minimally cover $X_{328}$
Curves that minimally cover $X_{328}$ and have infinitely many rational points.
Model $y^2 = x^3 + 8x$
Info about rational points $X_{328}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 74098773x - 2202231376821$, with conductor $76614$
Generic density of odd order reductions $12833/57344$