## The modular curve $X_{693}$

Curve name $X_{693}$
Index $96$
Level $32$
Genus $5$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 23 & 19 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 25 & 4 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{23}$ $8$ $24$ $X_{69}$ $16$ $48$ $X_{324}$
Meaning/Special name
Chosen covering $X_{324}$
Curves that $X_{693}$ minimally covers $X_{324}$
Curves that minimally cover $X_{693}$
Curves that minimally cover $X_{693}$ and have infinitely many rational points.
Model $y^2 = x^3 - 2x$$w^2 = -8x^2y - 8xy - 2y^3$
 Rational point Image on the $j$-line $(0 : 0 : 1 : 0)$ Singular $(0 : 0 : 0 : 1)$ Singular
Elliptic curve whose $2$-adic image is the subgroup None