## The modular curve $X_{376}$

Curve name $X_{376}$
Index $48$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 12 & 15 \end{matrix}\right], \left[ \begin{matrix} 3 & 9 \\ 4 & 13 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 12 & 15 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{27}$ $8$ $24$ $X_{94}$
Meaning/Special name
Chosen covering $X_{94}$
Curves that $X_{376}$ minimally covers $X_{94}$, $X_{150}$, $X_{153}$
Curves that minimally cover $X_{376}$
Curves that minimally cover $X_{376}$ and have infinitely many rational points.
Model $y^2 = x^6 - 5x^4 - 5x^2 + 1$
 Rational point Image on the $j$-line $(1 : -1 : 0)$ $1728 \,\,(\text{CM by }-4)$ $(1 : 1 : 0)$ $1728 \,\,(\text{CM by }-4)$ $(0 : -1 : 1)$ $1728 \,\,(\text{CM by }-4)$ $(0 : 1 : 1)$ $1728 \,\,(\text{CM by }-4)$
Elliptic curve whose $2$-adic image is the subgroup None