The modular curve $X_{9}$

Curve name $X_{9}$
Index $6$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
Meaning/Special name Elliptic curves that acquire a cyclic $4$-isogeny over $\mathbb{Q}(i)$
Chosen covering $X_{6}$
Curves that $X_{9}$ minimally covers $X_{6}$
Curves that minimally cover $X_{9}$ $X_{24}$, $X_{37}$, $X_{9a}$, $X_{9b}$, $X_{9c}$, $X_{9d}$
Curves that minimally cover $X_{9}$ and have infinitely many rational points. $X_{24}$, $X_{37}$, $X_{9a}$, $X_{9b}$, $X_{9c}$, $X_{9d}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{9}) = \mathbb{Q}(f_{9}), f_{6} = f_{9}^{2} + 48\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 12x - 11$, with conductor $180$
Generic density of odd order reductions $83/336$

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