## The modular curve $X_{200}$

Curve name $X_{200}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{62}$
Curves that $X_{200}$ minimally covers $X_{62}$, $X_{98}$, $X_{101}$
Curves that minimally cover $X_{200}$ $X_{451}$, $X_{455}$, $X_{456}$, $X_{462}$, $X_{200a}$, $X_{200b}$, $X_{200c}$, $X_{200d}$, $X_{200e}$, $X_{200f}$, $X_{200g}$, $X_{200h}$
Curves that minimally cover $X_{200}$ and have infinitely many rational points. $X_{200a}$, $X_{200b}$, $X_{200c}$, $X_{200d}$, $X_{200e}$, $X_{200f}$, $X_{200g}$, $X_{200h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{200}) = \mathbb{Q}(f_{200}), f_{62} = \frac{f_{200}^{2} - \frac{1}{2}}{f_{200}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 936x - 3668$, with conductor $126$
Generic density of odd order reductions $193/1792$