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

Curve name $X_{194k}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25h}$
Meaning/Special name
Chosen covering $X_{194}$
Curves that $X_{194k}$ minimally covers
Curves that minimally cover $X_{194k}$
Curves that minimally cover $X_{194k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1769472t^{16} - 3538944t^{14} - 1327104t^{12} + 221184t^{10} - 1589760t^{8} + 13824t^{6} - 5184t^{4} - 864t^{2} - 27$ $B(t) = -905969664t^{24} - 2717908992t^{22} - 2378170368t^{20} - 396361728t^{18} + 1985347584t^{16} + 1847328768t^{14} - 229146624t^{12} + 115458048t^{10} + 7755264t^{8} - 96768t^{6} - 36288t^{4} - 2592t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 7220x - 224143$, with conductor $510$
Generic density of odd order reductions $19/336$