## The modular curve $X_{377}$

Curve name $X_{377}$
Index $48$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 4 \\ 10 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 7 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{97}$
Meaning/Special name
Chosen covering $X_{97}$
Curves that $X_{377}$ minimally covers $X_{97}$
Curves that minimally cover $X_{377}$
Curves that minimally cover $X_{377}$ and have infinitely many rational points.
Model $y^2 = x^5 - 6x^3 + x$
 Rational point Image on the $j$-line $(1 : 0 : 0)$ $1728 \,\,(\text{CM by }-4)$ $(-1 : -2 : 1)$ $1728 \,\,(\text{CM by }-4)$ $(-1 : 2 : 1)$ $1728 \,\,(\text{CM by }-4)$ $(0 : 0 : 1)$ $1728 \,\,(\text{CM by }-4)$
Elliptic curve whose $2$-adic image is the subgroup None