## The modular curve $X_{199}$

Curve name $X_{199}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{199}$ minimally covers $X_{75}$, $X_{84}$, $X_{85}$
Curves that minimally cover $X_{199}$ $X_{466}$, $X_{477}$, $X_{199a}$, $X_{199b}$, $X_{199c}$, $X_{199d}$, $X_{199e}$, $X_{199f}$, $X_{199g}$, $X_{199h}$
Curves that minimally cover $X_{199}$ and have infinitely many rational points. $X_{199a}$, $X_{199b}$, $X_{199c}$, $X_{199d}$, $X_{199e}$, $X_{199f}$, $X_{199g}$, $X_{199h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{199}) = \mathbb{Q}(f_{199}), f_{75} = \frac{f_{199}}{f_{199}^{2} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 14796x + 835434$, with conductor $306$
Generic density of odd order reductions $635/5376$