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

Curve name $X_{194i}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
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 $X_{194i}$ minimally covers
Curves that minimally cover $X_{194i}$
Curves that minimally cover $X_{194i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1811939328t^{24} - 9059696640t^{22} - 16533946368t^{20} - 12796821504t^{18} - 4862509056t^{16} - 4600627200t^{14} - 3791978496t^{12} - 287539200t^{10} - 18994176t^{8} - 3124224t^{6} - 252288t^{4} - 8640t^{2} - 108$ $B(t) = 29686813949952t^{36} + 222651104624640t^{34} + 684652146720768t^{32} + 1098412116148224t^{30} + 897562265518080t^{28} + 82798379532288t^{26} - 629714146295808t^{24} - 639078248742912t^{22} - 231497898393600t^{20} - 25209058885632t^{18} - 14468618649600t^{16} - 2496399409152t^{14} - 153738805248t^{12} + 1263403008t^{10} + 855982080t^{8} + 65470464t^{6} + 2550528t^{4} + 51840t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 33385376x + 69542885376$, with conductor $69360$
Generic density of odd order reductions $109/896$