## The modular curve $X_{217}$

Curve name $X_{217}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{75}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{217}$ minimally covers $X_{75}$, $X_{120}$, $X_{122}$
Curves that minimally cover $X_{217}$ $X_{466}$, $X_{467}$, $X_{477}$, $X_{481}$, $X_{217a}$, $X_{217b}$, $X_{217c}$, $X_{217d}$, $X_{217e}$, $X_{217f}$, $X_{217g}$, $X_{217h}$
Curves that minimally cover $X_{217}$ and have infinitely many rational points. $X_{217a}$, $X_{217b}$, $X_{217c}$, $X_{217d}$, $X_{217e}$, $X_{217f}$, $X_{217g}$, $X_{217h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{217}) = \mathbb{Q}(f_{217}), f_{75} = \frac{f_{217}}{f_{217}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 3474x - 31010$, with conductor $126$
Generic density of odd order reductions $193/1792$