## The modular curve $X_{649}$

Curve name $X_{649}$
Index $96$
Level $32$
Genus $3$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 29 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 31 & 27 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 31 & 31 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{23}$ $8$ $24$ $X_{69}$ $16$ $48$ $X_{323}$
Meaning/Special name
Chosen covering $X_{323}$
Curves that $X_{649}$ minimally covers $X_{323}$
Curves that minimally cover $X_{649}$
Curves that minimally cover $X_{649}$ and have infinitely many rational points.
Model $x^4 - x^2y^2 - 2x^2z^2 - y^3z + 2yz^3 = 0$
 Rational point Image on the $j$-line $(-2 : -4 : 1)$ $\frac{777228872334890625}{60523872256}$ $(0 : 1 : 0)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(1 : 1 : 0)$ $-3375 \,\,(\text{CM by }-7)$ $(2 : -4 : 1)$ $\frac{777228872334890625}{60523872256}$ $(-1 : 1 : 0)$ $-3375 \,\,(\text{CM by }-7)$
Comments on finding rational points This curve is isomorphic to $X_{619}$.
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 325630x - 71434867$, with conductor $17918$
Generic density of odd order reductions $410657/1835008$