The modular curve $X_{32k}$

Curve name $X_{32k}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 3 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{32}$
Curves that $X_{32k}$ minimally covers
Curves that minimally cover $X_{32k}$
Curves that minimally cover $X_{32k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{4} - 1728t^{2} - 1728\] \[B(t) = -432t^{6} - 10368t^{4} - 51840t^{2} + 27648\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 1921x - 31775$, with conductor $960$
Generic density of odd order reductions $289/1792$

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