The modular curve $X_{253}$

Curve name $X_{253}$
Index $48$
Level $8$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 5 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 6 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{26}$
Meaning/Special name
Chosen covering $X_{73}$
Curves that $X_{253}$ minimally covers $X_{55}$, $X_{73}$
Curves that minimally cover $X_{253}$ $X_{537}$, $X_{538}$, $X_{596}$, $X_{597}$
Curves that minimally cover $X_{253}$ and have infinitely many rational points.
Model \[y^2 = x^3 + x\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[0 \,\,(\text{CM by }-3)\]
$(0 : 0 : 1)$ \[0 \,\,(\text{CM by }-3)\]
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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