The modular curve $X_{80}$

Curve name $X_{80}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{23}$
Meaning/Special name
Chosen covering $X_{23}$
Curves that $X_{80}$ minimally covers $X_{23}$, $X_{39}$, $X_{41}$
Curves that minimally cover $X_{80}$ $X_{224}$, $X_{261}$, $X_{262}$, $X_{263}$, $X_{264}$, $X_{366}$, $X_{400}$
Curves that minimally cover $X_{80}$ and have infinitely many rational points. $X_{224}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{80}) = \mathbb{Q}(f_{80}), f_{23} = \frac{f_{80}}{f_{80}^{2} + \frac{1}{8}}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 183x - 765$, with conductor $5376$
Generic density of odd order reductions $401/1792$

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