## The modular curve $X_{302}$

Curve name $X_{302}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 13 & 7 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 10 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{63}$
Meaning/Special name
Chosen covering $X_{63}$
Curves that $X_{302}$ minimally covers $X_{63}$, $X_{105}$, $X_{166}$
Curves that minimally cover $X_{302}$
Curves that minimally cover $X_{302}$ and have infinitely many rational points.
Model $y^2 = x^3 + x^2 - 13x - 21$
Info about rational points $X_{302}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 7859386910x - 268185327947856$, with conductor $30631008$
Generic density of odd order reductions $42979/172032$