The modular curve $X_{199d}$

Curve name $X_{199d}$
Index $96$
Level $16$
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 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{199}$
Meaning/Special name
Chosen covering $X_{199}$
Curves that $X_{199d}$ minimally covers
Curves that minimally cover $X_{199d}$
Curves that minimally cover $X_{199d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{24} + 47520t^{22} + 870048t^{20} - 1012608t^{18} - 69201216t^{16} - 219663360t^{14} - 394343424t^{12} - 878653440t^{10} - 1107219456t^{8} - 64806912t^{6} + 222732288t^{4} + 48660480t^{2} - 442368\] \[B(t) = -432t^{36} - 461376t^{34} + 3716928t^{32} + 510879744t^{30} + 7607043072t^{28} + 35131428864t^{26} + 38478827520t^{24} + 466915000320t^{22} + 3442233729024t^{20} + 9889733836800t^{18} + 13768934916096t^{16} + 7470640005120t^{14} + 2462644961280t^{12} + 8993645789184t^{10} + 7789612105728t^{8} + 2092563431424t^{6} + 60898148352t^{4} - 30236737536t^{2} - 113246208\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 30407809x + 77498755105$, with conductor $55488$
Generic density of odd order reductions $109/896$

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