## The modular curve $X_{204}$

Curve name $X_{204}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
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 $X_{204}$ minimally covers $X_{62}$, $X_{87}$, $X_{98}$
Curves that minimally cover $X_{204}$ $X_{448}$, $X_{455}$, $X_{456}$, $X_{464}$, $X_{204a}$, $X_{204b}$, $X_{204c}$, $X_{204d}$, $X_{204e}$, $X_{204f}$, $X_{204g}$, $X_{204h}$
Curves that minimally cover $X_{204}$ and have infinitely many rational points. $X_{204a}$, $X_{204b}$, $X_{204c}$, $X_{204d}$, $X_{204e}$, $X_{204f}$, $X_{204g}$, $X_{204h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{204}) = \mathbb{Q}(f_{204}), f_{62} = \frac{2f_{204}^{2} + 8}{f_{204}^{2} + 4f_{204} - 4}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 608x - 5712$, with conductor $600$
Generic density of odd order reductions $635/5376$