## The modular curve $X_{197}$

Curve name $X_{197}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
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 $X_{197}$ minimally covers $X_{75}$, $X_{86}$, $X_{102}$
Curves that minimally cover $X_{197}$ $X_{467}$, $X_{481}$, $X_{197a}$, $X_{197b}$, $X_{197c}$, $X_{197d}$, $X_{197e}$, $X_{197f}$, $X_{197g}$, $X_{197h}$
Curves that minimally cover $X_{197}$ and have infinitely many rational points. $X_{197a}$, $X_{197b}$, $X_{197c}$, $X_{197d}$, $X_{197e}$, $X_{197f}$, $X_{197g}$, $X_{197h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{197}) = \mathbb{Q}(f_{197}), f_{75} = \frac{\frac{1}{4}f_{197}^{2} + f_{197} - 1}{f_{197}^{2} + 4}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 392x - 21712$, with conductor $600$
Generic density of odd order reductions $635/5376$