The modular curve $X_{200h}$

Curve name $X_{200h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{200}$
Curves that $X_{200h}$ minimally covers
Curves that minimally cover $X_{200h}$
Curves that minimally cover $X_{200h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -442368t^{24} + 4423680t^{22} - 42688512t^{20} + 237330432t^{18} - 629462016t^{16} + 832204800t^{14} - 600044544t^{12} + 208051200t^{10} - 39341376t^{8} + 3708288t^{6} - 166752t^{4} + 4320t^{2} - 108\] \[B(t) = 113246208t^{36} - 1698693120t^{34} - 3822059520t^{32} + 151976411136t^{30} - 1011826556928t^{28} + 3649896972288t^{26} - 8480605077504t^{24} + 12985617088512t^{22} - 12688948297728t^{20} + 7860634288128t^{18} - 3172237074432t^{16} + 811601068032t^{14} - 132509454336t^{12} + 14257410048t^{10} - 988111872t^{8} + 37103616t^{6} - 233280t^{4} - 25920t^{2} + 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 326209x - 25271231$, with conductor $9408$
Generic density of odd order reductions $109/896$

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