## The modular curve $X_{208}$

Curve name $X_{208}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $24$ $X_{96}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{208}$ minimally covers $X_{96}$, $X_{121}$, $X_{122}$
Curves that minimally cover $X_{208}$ $X_{468}$, $X_{475}$, $X_{482}$, $X_{486}$, $X_{208a}$, $X_{208b}$, $X_{208c}$, $X_{208d}$, $X_{208e}$, $X_{208f}$, $X_{208g}$, $X_{208h}$, $X_{208i}$, $X_{208j}$, $X_{208k}$, $X_{208l}$
Curves that minimally cover $X_{208}$ and have infinitely many rational points. $X_{208a}$, $X_{208b}$, $X_{208c}$, $X_{208d}$, $X_{208e}$, $X_{208f}$, $X_{208g}$, $X_{208h}$, $X_{208i}$, $X_{208j}$, $X_{208k}$, $X_{208l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{208}) = \mathbb{Q}(f_{208}), f_{96} = f_{208}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 12240x - 518144$, with conductor $1530$
Generic density of odd order reductions $25/224$