The modular curve $X_{7}$

Curve name $X_{7}$
Index $4$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $1$ $X_{1}$
Meaning/Special name Elliptic curves with $a_{p}(E) \equiv 0 \pmod{4}$ if $p$ is inert in $\mathbb{Q}(\sqrt{\Delta})$
Chosen covering $X_{1}$
Curves that $X_{7}$ minimally covers $X_{1}$
Curves that minimally cover $X_{7}$ $X_{20}$, $X_{21}$, $X_{22}$, $X_{26}$, $X_{55}$
Curves that minimally cover $X_{7}$ and have infinitely many rational points. $X_{20}$, $X_{22}$, $X_{26}$, $X_{55}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{7}) = \mathbb{Q}(f_{7}), f_{1} = \frac{32f_{7} - 4}{f_{7}^{4}}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 12x + 20$, with conductor $216$
Generic density of odd order reductions $179/336$

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