## The modular curve $X_{223}$

Curve name $X_{223}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{102}$
Meaning/Special name
Chosen covering $X_{102}$
Curves that $X_{223}$ minimally covers $X_{102}$, $X_{117}$, $X_{121}$
Curves that minimally cover $X_{223}$ $X_{474}$, $X_{475}$, $X_{223a}$, $X_{223b}$, $X_{223c}$, $X_{223d}$, $X_{223e}$, $X_{223f}$, $X_{223g}$, $X_{223h}$
Curves that minimally cover $X_{223}$ and have infinitely many rational points. $X_{223a}$, $X_{223b}$, $X_{223c}$, $X_{223d}$, $X_{223e}$, $X_{223f}$, $X_{223g}$, $X_{223h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{223}) = \mathbb{Q}(f_{223}), f_{102} = 8f_{223}^{2} + 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 306x - 1836$, with conductor $306$
Generic density of odd order reductions $635/5376$