## The modular curve $X_{199f}$

Curve name $X_{199f}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$
Meaning/Special name
Chosen covering $X_{199}$
Curves that $X_{199f}$ minimally covers
Curves that minimally cover $X_{199f}$
Curves that minimally cover $X_{199f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} + 12528t^{14} - 79056t^{12} - 257472t^{10} - 306720t^{8} - 1029888t^{6} - 1264896t^{4} + 801792t^{2} - 6912$ $B(t) = 54t^{24} + 55728t^{22} - 2494800t^{20} + 1100736t^{18} + 4740768t^{16} - 120310272t^{14} - 404006400t^{12} - 481241088t^{10} + 75852288t^{8} + 70447104t^{6} - 638668800t^{4} + 57065472t^{2} + 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 1644x - 30942$, with conductor $102$
Generic density of odd order reductions $53/896$