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
LevelIndex of imageCorresponding 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 \[y^2 = x^3 - 4x\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[ \infty \]
$(-2 : 0 : 1)$ \[1728 \,\,(\text{CM by }-4)\]
$(0 : 0 : 1)$ \[ \infty \]
$(2 : 0 : 1)$ \[1728 \,\,(\text{CM by }-4)\]
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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