The modular curve $X_{199a}$

Curve name $X_{199a}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 7 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 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 $X_{199a}$ minimally covers
Curves that minimally cover $X_{199a}$
Curves that minimally cover $X_{199a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{24} + 11880t^{22} + 217512t^{20} - 253152t^{18} - 17300304t^{16} - 54915840t^{14} - 98585856t^{12} - 219663360t^{10} - 276804864t^{8} - 16201728t^{6} + 55683072t^{4} + 12165120t^{2} - 110592\] \[B(t) = -54t^{36} - 57672t^{34} + 464616t^{32} + 63859968t^{30} + 950880384t^{28} + 4391428608t^{26} + 4809853440t^{24} + 58364375040t^{22} + 430279216128t^{20} + 1236216729600t^{18} + 1721116864512t^{16} + 933830000640t^{14} + 307830620160t^{12} + 1124205723648t^{10} + 973701513216t^{8} + 261570428928t^{6} + 7612268544t^{4} - 3779592192t^{2} - 14155776\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 7601952x + 9683543412$, with conductor $13872$
Generic density of odd order reductions $299/2688$

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