## The modular curve $X_{219}$

Curve name $X_{219}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{85}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{219}$ minimally covers $X_{85}$, $X_{117}$, $X_{119}$
Curves that minimally cover $X_{219}$ $X_{473}$, $X_{483}$, $X_{219a}$, $X_{219b}$, $X_{219c}$, $X_{219d}$, $X_{219e}$, $X_{219f}$, $X_{219g}$, $X_{219h}$
Curves that minimally cover $X_{219}$ and have infinitely many rational points. $X_{219a}$, $X_{219b}$, $X_{219c}$, $X_{219d}$, $X_{219e}$, $X_{219f}$, $X_{219g}$, $X_{219h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{219}) = \mathbb{Q}(f_{219}), f_{85} = \frac{f_{219}^{2} + 2f_{219} - 1}{f_{219}^{2} + 1}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 1608x + 24288$, with conductor $600$
Generic density of odd order reductions $635/5376$