The modular curve $X_{70}$

Curve name $X_{70}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{45}$
Curves that $X_{70}$ minimally covers $X_{45}$, $X_{47}$, $X_{49}$
Curves that minimally cover $X_{70}$ $X_{257}$, $X_{260}$, $X_{325}$, $X_{326}$
Curves that minimally cover $X_{70}$ and have infinitely many rational points. $X_{326}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{70}) = \mathbb{Q}(f_{70}), f_{45} = \frac{8f_{70}}{f_{70}^{2} + 2}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 513x - 5103$, with conductor $2400$
Generic density of odd order reductions $1343/5376$

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