The modular curve $X_{197h}$

Curve name $X_{197h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{197}$
Curves that $X_{197h}$ minimally covers
Curves that minimally cover $X_{197h}$
Curves that minimally cover $X_{197h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 1269t^{16} - 50976t^{15} - 273888t^{14} + 584064t^{13} - 2058048t^{12} - 13754880t^{11} + 11349504t^{10} - 67627008t^{9} + 92634624t^{8} + 270508032t^{7} + 181592064t^{6} + 880312320t^{5} - 526860288t^{4} - 598081536t^{3} - 1121845248t^{2} + 835190784t + 83165184\] \[B(t) = 127386t^{24} + 1008288t^{23} - 5051808t^{22} + 36408960t^{21} + 505528128t^{20} + 109444608t^{19} - 543158784t^{18} + 10318067712t^{17} - 30131108352t^{16} + 15869952000t^{15} + 139933827072t^{14} - 102893027328t^{13} + 1801903767552t^{12} + 411572109312t^{11} + 2238941233152t^{10} - 1015676928000t^{9} - 7713563738112t^{8} - 10565701337088t^{7} - 2224778379264t^{6} - 1793140457472t^{5} + 33130291396608t^{4} - 9544390410240t^{3} - 5297204625408t^{2} - 4229066391552t + 2137182437376\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 16x + 180$, with conductor $48$
Generic density of odd order reductions $53/896$

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