## The modular curve $X_{404}$

Curve name $X_{404}$
Index $48$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $24$ $X_{96}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{404}$ minimally covers $X_{96}$
Curves that minimally cover $X_{404}$
Curves that minimally cover $X_{404}$ and have infinitely many rational points.
Model $y^2 = x^5 - x$
Info about rational points
 Rational point Image on the $j$-line $(1 : 0 : 0)$ $\infty$ $(-1 : 0 : 1)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(1 : 0 : 1)$ $\infty$
Comments on finding rational points The rank of the Jacobian is 0. We use the method of Chabauty.
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A