The modular curve $X_{193j}$

Curve name $X_{193j}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 6 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{25}$
$8$ $48$ $X_{193}$
Meaning/Special name
Chosen covering $X_{193}$
Curves that $X_{193j}$ minimally covers
Curves that minimally cover $X_{193j}$
Curves that minimally cover $X_{193j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -113246208t^{22} + 283115520t^{20} - 205258752t^{18} + 42467328t^{16} - 99975168t^{14} + 49102848t^{12} - 6248448t^{10} + 165888t^{8} - 50112t^{6} + 4320t^{4} - 108t^{2}\] \[B(t) = 463856467968t^{33} - 1739461754880t^{31} + 2348273369088t^{29} - 1384321646592t^{27} - 614247432192t^{25} + 1651129712640t^{23} - 779473649664t^{21} + 164348559360t^{19} - 41087139840t^{17} + 12179275776t^{15} - 1612431360t^{13} + 37490688t^{11} + 5280768t^{9} - 559872t^{7} + 25920t^{5} - 432t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 616332x - 109225744$, with conductor $20160$
Generic density of odd order reductions $11/112$

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