The modular curve $X_{199h}$

Curve name $X_{199h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{199}$
Curves that $X_{199h}$ minimally covers
Curves that minimally cover $X_{199h}$
Curves that minimally cover $X_{199h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{16} + 50112t^{14} - 316224t^{12} - 1029888t^{10} - 1226880t^{8} - 4119552t^{6} - 5059584t^{4} + 3207168t^{2} - 27648\] \[B(t) = 432t^{24} + 445824t^{22} - 19958400t^{20} + 8805888t^{18} + 37926144t^{16} - 962482176t^{14} - 3232051200t^{12} - 3849928704t^{10} + 606818304t^{8} + 563576832t^{6} - 5109350400t^{4} + 456523776t^{2} + 1769472\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 105217x - 15737087$, with conductor $3264$
Generic density of odd order reductions $109/896$

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