The modular curve $X_{178}$

Curve name $X_{178}$
Index $32$
Level $32$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 9 & 31 \\ 1 & 2 \end{matrix}\right], \left[ \begin{matrix} 23 & 19 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $1$ $X_{1}$
$4$ $4$ $X_{7}$
$8$ $8$ $X_{22}$
$16$ $16$ $X_{56}$
Meaning/Special name
Chosen covering $X_{56}$
Curves that $X_{178}$ minimally covers $X_{56}$
Curves that minimally cover $X_{178}$ $X_{718}$
Curves that minimally cover $X_{178}$ and have infinitely many rational points.
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{178}) = \mathbb{Q}(f_{178}), f_{56} = \frac{3f_{178}^{2} + 3}{f_{178}^{2} + 2f_{178} - 1}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 120x - 512$, with conductor $20736$
Generic density of odd order reductions $977931/1835008$

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