The modular curve $X_{237}$

Curve name $X_{237}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 11 & 11 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{26}$
$8$ $24$ $X_{83}$
Meaning/Special name
Chosen covering $X_{83}$
Curves that $X_{237}$ minimally covers $X_{83}$, $X_{106}$, $X_{107}$
Curves that minimally cover $X_{237}$
Curves that minimally cover $X_{237}$ and have infinitely many rational points.
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{237}) = \mathbb{Q}(f_{237}), f_{83} = \frac{f_{237}}{f_{237}^{2} + 2}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 221950x + 40248384$, with conductor $147712$
Generic density of odd order reductions $45667/172032$

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