## The modular curve $X_{654}$

Curve name $X_{654}$
Index $96$
Level $32$
Genus $3$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 20 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 28 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 20 & 13 \end{matrix}\right]$
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 $X_{654}$ minimally covers $X_{324}$
Curves that minimally cover $X_{654}$
Curves that minimally cover $X_{654}$ and have infinitely many rational points.
Model $-x^3y + 2xy^3 - z^4 = 0$
 Rational point Image on the $j$-line $(1 : 1 : 1)$ $-3375 \,\,(\text{CM by }-7)$ $(0 : 1 : 0)$ $\infty$ $(1 : 0 : 0)$ $\infty$ $(-1 : -1 : 1)$ $-3375 \,\,(\text{CM by }-7)$
Comments on finding rational points This curve is isomorphic to $X_{628}$.
Elliptic curve whose $2$-adic image is the subgroup None