## The modular curve $X_{194}$

Curve name $X_{194}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{194}$ minimally covers $X_{96}$, $X_{98}$, $X_{99}$
Curves that minimally cover $X_{194}$ $X_{451}$, $X_{460}$, $X_{464}$, $X_{465}$, $X_{468}$, $X_{469}$, $X_{194a}$, $X_{194b}$, $X_{194c}$, $X_{194d}$, $X_{194e}$, $X_{194f}$, $X_{194g}$, $X_{194h}$, $X_{194i}$, $X_{194j}$, $X_{194k}$, $X_{194l}$
Curves that minimally cover $X_{194}$ and have infinitely many rational points. $X_{194a}$, $X_{194b}$, $X_{194c}$, $X_{194d}$, $X_{194e}$, $X_{194f}$, $X_{194g}$, $X_{194h}$, $X_{194i}$, $X_{194j}$, $X_{194k}$, $X_{194l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{194}) = \mathbb{Q}(f_{194}), f_{96} = \frac{f_{194}^{2} + \frac{1}{4}}{f_{194}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 64980x + 5986876$, with conductor $1530$
Generic density of odd order reductions $25/224$