The modular curve $X_{193b}$

Curve name	$X_{193b}$
Index	$96$
Level	$16$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{193}$
Meaning/Special name
Chosen covering	$X_{193}$
Curves that $X_{193b}$ minimally covers
Curves that minimally cover $X_{193b}$
Curves that minimally cover $X_{193b}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -28311552t^{22} + 70778880t^{20} - 51314688t^{18} + 10616832t^{16} - 24993792t^{14} + 12275712t^{12} - 1562112t^{10} + 41472t^{8} - 12528t^{6} + 1080t^{4} - 27t^{2}$$ $$B(t) = 57982058496t^{33} - 217432719360t^{31} + 293534171136t^{29} - 173040205824t^{27} - 76780929024t^{25} + 206391214080t^{23} - 97434206208t^{21} + 20543569920t^{19} - 5135892480t^{17} + 1522409472t^{15} - 201553920t^{13} + 4686336t^{11} + 660096t^{9} - 69984t^{7} + 3240t^{5} - 54t^{3}$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 + xy = x^3 - x^2 - 9630x - 210924$, with conductor $630$
Generic density of odd order reductions	$271/2688$

Back to the 2-adic image homepage.