## The modular curve $X_{228}$

Curve name $X_{228}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{84}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{228}$ minimally covers $X_{84}$, $X_{117}$, $X_{122}$
Curves that minimally cover $X_{228}$ $X_{466}$, $X_{487}$, $X_{228a}$, $X_{228b}$, $X_{228c}$, $X_{228d}$, $X_{228e}$, $X_{228f}$, $X_{228g}$, $X_{228h}$
Curves that minimally cover $X_{228}$ and have infinitely many rational points. $X_{228a}$, $X_{228b}$, $X_{228c}$, $X_{228d}$, $X_{228e}$, $X_{228f}$, $X_{228g}$, $X_{228h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{228}) = \mathbb{Q}(f_{228}), f_{84} = \frac{f_{228}^{2} - 8}{f_{228}^{2} + 8f_{228} + 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 17x + 38$, with conductor $600$
Generic density of odd order reductions $635/5376$