## The modular curve $X_{501}$

Curve name $X_{501}$
Index $96$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 15 \end{matrix}\right], \left[ \begin{matrix} 7 & 12 \\ 0 & 11 \end{matrix}\right], \left[ \begin{matrix} 3 & 12 \\ 12 & 11 \end{matrix}\right], \left[ \begin{matrix} 3 & 12 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 10 & 15 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{58}$ $8$ $48$ $X_{189}$
Meaning/Special name
Chosen covering $X_{189}$
Curves that $X_{501}$ minimally covers $X_{189}$
Curves that minimally cover $X_{501}$
Curves that minimally cover $X_{501}$ and have infinitely many rational points.
Model $y^2 + (x^3 + x^2 + x + 1)y = -x^5 - x^4 - x^3 - x^2$
 Rational point Image on the $j$-line $(1 : -1 : 0)$ $\infty$ $(1 : 0 : 0)$ $\infty$ $(-1 : 0 : 1)$ $\infty$ $(0 : -1 : 1)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(1 : -2 : 1)$ $\infty$
Elliptic curve whose $2$-adic image is the subgroup None