## The modular curve $X_{195}$

Curve name $X_{195}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{27}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{195}$ minimally covers $X_{84}$, $X_{92}$, $X_{102}$
Curves that minimally cover $X_{195}$ $X_{445}$, $X_{453}$, $X_{479}$, $X_{487}$, $X_{195a}$, $X_{195b}$, $X_{195c}$, $X_{195d}$, $X_{195e}$, $X_{195f}$, $X_{195g}$, $X_{195h}$, $X_{195i}$, $X_{195j}$, $X_{195k}$, $X_{195l}$
Curves that minimally cover $X_{195}$ and have infinitely many rational points. $X_{195a}$, $X_{195b}$, $X_{195c}$, $X_{195d}$, $X_{195e}$, $X_{195f}$, $X_{195g}$, $X_{195h}$, $X_{195i}$, $X_{195j}$, $X_{195k}$, $X_{195l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{195}) = \mathbb{Q}(f_{195}), f_{84} = \frac{f_{195}^{2} - \frac{1}{2}}{f_{195}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + 1714x + 14685$, with conductor $735$
Generic density of odd order reductions $25/224$