## The modular curve $X_{203}$

Curve name $X_{203}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right]$
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 $X_{203}$ minimally covers $X_{62}$, $X_{99}$, $X_{101}$
Curves that minimally cover $X_{203}$ $X_{442}$, $X_{451}$, $X_{462}$, $X_{463}$, $X_{203a}$, $X_{203b}$, $X_{203c}$, $X_{203d}$, $X_{203e}$, $X_{203f}$, $X_{203g}$, $X_{203h}$
Curves that minimally cover $X_{203}$ and have infinitely many rational points. $X_{203a}$, $X_{203b}$, $X_{203c}$, $X_{203d}$, $X_{203e}$, $X_{203f}$, $X_{203g}$, $X_{203h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{203}) = \mathbb{Q}(f_{203}), f_{62} = \frac{f_{203}}{f_{203}^{2} + \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 15606x + 754272$, with conductor $306$
Generic density of odd order reductions $635/5376$