The modular curve $X_{8b}$

Curve name $X_{8b}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
Meaning/Special name
Chosen covering $X_{8}$
Curves that $X_{8b}$ minimally covers
Curves that minimally cover $X_{8b}$
Curves that minimally cover $X_{8b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{6} - 108t^{4} + 540t^{2} - 324\] \[B(t) = 432t^{9} - 5184t^{7} + 12960t^{5} - 12096t^{3} + 3888t\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 12546x + 173047$, with conductor $1089$
Generic density of odd order reductions $9/56$

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