The modular curve $X_{218}$

Curve name $X_{218}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 10 & 15 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
$8$ $24$ $X_{81}$
Meaning/Special name
Chosen covering $X_{81}$
Curves that $X_{218}$ minimally covers $X_{81}$, $X_{110}$, $X_{111}$
Curves that minimally cover $X_{218}$
Curves that minimally cover $X_{218}$ and have infinitely many rational points.
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{218}) = \mathbb{Q}(f_{218}), f_{81} = \frac{f_{218}}{f_{218}^{2} + \frac{1}{8}}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 3551240x + 2575835648$, with conductor $147712$
Generic density of odd order reductions $45667/172032$

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