## The modular curve $X_{324}$

Curve name $X_{324}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
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 $X_{324}$ minimally covers $X_{69}$, $X_{113}$, $X_{156}$
Curves that minimally cover $X_{324}$ $X_{568}$, $X_{569}$, $X_{570}$, $X_{575}$, $X_{654}$, $X_{693}$, $X_{694}$, $X_{712}$
Curves that minimally cover $X_{324}$ and have infinitely many rational points.
Model $y^2 = x^3 - 2x$
Info about rational points $X_{324}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 3503663358x + 159400872404283$, with conductor $2665869738$
Generic density of odd order reductions $12833/57344$