## The modular curve $X_{193}$

Curve name $X_{193}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name $\tilde{X}_{1}(2,8)$
Chosen covering $X_{96}$
Curves that $X_{193}$ minimally covers $X_{96}$, $X_{98}$, $X_{102}$
Curves that minimally cover $X_{193}$ $X_{456}$, $X_{465}$, $X_{470}$, $X_{475}$, $X_{507}$, $X_{508}$, $X_{509}$, $X_{510}$, $X_{193a}$, $X_{193b}$, $X_{193c}$, $X_{193d}$, $X_{193e}$, $X_{193f}$, $X_{193g}$, $X_{193h}$, $X_{193i}$, $X_{193j}$, $X_{193k}$, $X_{193l}$, $X_{193m}$, $X_{193n}$
Curves that minimally cover $X_{193}$ and have infinitely many rational points. $X_{193a}$, $X_{193b}$, $X_{193c}$, $X_{193d}$, $X_{193e}$, $X_{193f}$, $X_{193g}$, $X_{193h}$, $X_{193i}$, $X_{193j}$, $X_{193k}$, $X_{193l}$, $X_{193m}$, $X_{193n}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{193}) = \mathbb{Q}(f_{193}), f_{96} = \frac{f_{193}}{f_{193}^{2} - \frac{1}{4}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 129473x - 10527244$, with conductor $25410$
Generic density of odd order reductions $17/168$