## The modular curve $X_{402}$

Curve name $X_{402}$
Index $48$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 15 & 15 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 9 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 15 & 14 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{26}$ $8$ $24$ $X_{83}$
Meaning/Special name
Chosen covering $X_{83}$
Curves that $X_{402}$ minimally covers $X_{56}$, $X_{83}$, $X_{165}$, $X_{166}$
Curves that minimally cover $X_{402}$
Curves that minimally cover $X_{402}$ and have infinitely many rational points.
Model $y^2 = x^6 + 10x^4 - 20x^2 - 8$
 Rational point Image on the $j$-line $(1 : -1 : 0)$ $1728 \,\,(\text{CM by }-4)$ $(1 : 1 : 0)$ $1728 \,\,(\text{CM by }-4)$
Comments on finding rational points The rank of the Jacobian is 2. This curve admits a family of etale double covers that map to rank zero elliptic curves.
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A