The modular curve $X_{56}$

Curve name $X_{56}$
Index $16$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 13 & 11 \\ 1 & 6 \end{matrix}\right], \left[ \begin{matrix} 7 & 3 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $1$ $X_{1}$
$4$ $4$ $X_{7}$
$8$ $8$ $X_{22}$
Meaning/Special name
Chosen covering $X_{22}$
Curves that $X_{56}$ minimally covers $X_{22}$
Curves that minimally cover $X_{56}$ $X_{177}$, $X_{178}$, $X_{402}$, $X_{439}$
Curves that minimally cover $X_{56}$ and have infinitely many rational points. $X_{177}$, $X_{178}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{56}) = \mathbb{Q}(f_{56}), f_{22} = \frac{f_{56}^{2} + \frac{9}{2}}{f_{56}}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 17400x + 4787200$, with conductor $3628800$
Generic density of odd order reductions $91681/172032$

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