The modular curve $X_{252}$

Curve name $X_{252}$
Index $48$
Level $8$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 5 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{23}$
Meaning/Special name $X_{s}^{+}(8)$
Chosen covering $X_{69}$
Curves that $X_{252}$ minimally covers $X_{69}$
Curves that minimally cover $X_{252}$ $X_{536}$, $X_{539}$, $X_{542}$, $X_{543}$, $X_{568}$, $X_{569}$, $X_{576}$, $X_{581}$
Curves that minimally cover $X_{252}$ and have infinitely many rational points.
Model \[y^2 = x^3 - x\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[ \infty \]
$(-1 : 0 : 1)$ \[-3375 \,\,(\text{CM by }-7)\]
$(0 : 0 : 1)$ \[-3375 \,\,(\text{CM by }-7)\]
$(1 : 0 : 1)$ \[ \infty \]
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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