## The modular curve $X_{194b}$

Curve name $X_{194b}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{194}$
Meaning/Special name
Chosen covering $X_{194}$
Curves that $X_{194b}$ minimally covers
Curves that minimally cover $X_{194b}$
Curves that minimally cover $X_{194b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -113246208t^{22} - 283115520t^{20} - 205258752t^{18} - 42467328t^{16} - 99975168t^{14} - 49102848t^{12} - 6248448t^{10} - 165888t^{8} - 50112t^{6} - 4320t^{4} - 108t^{2}$ $B(t) = 463856467968t^{33} + 1739461754880t^{31} + 2348273369088t^{29} + 1384321646592t^{27} - 614247432192t^{25} - 1651129712640t^{23} - 779473649664t^{21} - 164348559360t^{19} - 41087139840t^{17} - 12179275776t^{15} - 1612431360t^{13} - 37490688t^{11} + 5280768t^{9} + 559872t^{7} + 25920t^{5} + 432t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 9872033x - 3668315937$, with conductor $196800$
Generic density of odd order reductions $299/2688$