## The modular curve $X_{53}$

Curve name $X_{53}$
Index $12$
Level $8$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{11}$
Curves that $X_{53}$ minimally covers $X_{11}$
Curves that minimally cover $X_{53}$ $X_{128}$, $X_{133}$, $X_{134}$, $X_{136}$, $X_{137}$, $X_{138}$, $X_{140}$, $X_{141}$, $X_{144}$, $X_{148}$
Curves that minimally cover $X_{53}$ and have infinitely many rational points.
Model $y^2 = x^3 - 4x$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(-2 : 0 : 1)$ $1728 \,\,(\text{CM by }-4)$ $(0 : 0 : 1)$ $\infty$ $(2 : 0 : 1)$ $1728 \,\,(\text{CM by }-4)$
Elliptic curve whose $2$-adic image is the subgroup None