The modular curve $X_{51}$

Curve name $X_{51}$
Index $12$
Level $8$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{12}$
Meaning/Special name
Chosen covering $X_{12}$
Curves that $X_{51}$ minimally covers $X_{12}$
Curves that minimally cover $X_{51}$ $X_{142}$, $X_{145}$, $X_{169}$, $X_{173}$, $X_{175}$, $X_{176}$
Curves that minimally cover $X_{51}$ and have infinitely many rational points.
Model \[y^2 = x^3 + 4x\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[ \infty \]
$(2 : 4 : 1)$ \[8000 \,\,(\text{CM by }-8)\]
$(0 : 0 : 1)$ \[ \infty \]
$(2 : -4 : 1)$ \[8000 \,\,(\text{CM by }-8)\]
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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