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

Curve name $X_{194f}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{194}$
Curves that $X_{194f}$ minimally covers
Curves that minimally cover $X_{194f}$
Curves that minimally cover $X_{194f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -7077888t^{16} - 14155776t^{14} - 5308416t^{12} + 884736t^{10} - 6359040t^{8} + 55296t^{6} - 20736t^{4} - 3456t^{2} - 108$ $B(t) = -7247757312t^{24} - 21743271936t^{22} - 19025362944t^{20} - 3170893824t^{18} + 15882780672t^{16} + 14778630144t^{14} - 1833172992t^{12} + 923664384t^{10} + 62042112t^{8} - 774144t^{6} - 290304t^{4} - 20736t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 462081x - 113374881$, with conductor $16320$
Generic density of odd order reductions $299/2688$