## The modular curve $X_{712}$

Curve name $X_{712}$
Index $96$
Level $32$
Genus $7$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 30 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 20 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 28 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 20 & 13 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{23}$ $8$ $24$ $X_{69}$ $16$ $48$ $X_{324}$
Meaning/Special name
Chosen covering $X_{324}$
Curves that $X_{712}$ minimally covers $X_{324}$
Curves that minimally cover $X_{712}$
Curves that minimally cover $X_{712}$ and have infinitely many rational points.
Model $y^2 = x^3 - 2x$$w^2 = 20x^2y - 2xy^3 - 8y$
 Rational point Image on the $j$-line $(0 : 0 : 1 : 0)$ $\infty$ $(0 : 0 : 0 : 1)$ Singular
Elliptic curve whose $2$-adic image is the subgroup None