## The modular curve $X_{202}$

Curve name $X_{202}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{78}$
Curves that $X_{202}$ minimally covers $X_{78}$, $X_{85}$, $X_{102}$
Curves that minimally cover $X_{202}$ $X_{471}$, $X_{472}$, $X_{473}$, $X_{474}$, $X_{202a}$, $X_{202b}$, $X_{202c}$, $X_{202d}$, $X_{202e}$, $X_{202f}$, $X_{202g}$, $X_{202h}$
Curves that minimally cover $X_{202}$ and have infinitely many rational points. $X_{202a}$, $X_{202b}$, $X_{202c}$, $X_{202d}$, $X_{202e}$, $X_{202f}$, $X_{202g}$, $X_{202h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{202}) = \mathbb{Q}(f_{202}), f_{78} = \frac{f_{202}^{2} - \frac{1}{2}}{f_{202}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 8226x + 286474$, with conductor $126$
Generic density of odd order reductions $193/1792$