## The modular curve $X_{213}$

Curve name $X_{213}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
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 $X_{213}$ minimally covers $X_{102}$, $X_{118}$, $X_{119}$
Curves that minimally cover $X_{213}$ $X_{470}$, $X_{472}$, $X_{491}$, $X_{492}$, $X_{213a}$, $X_{213b}$, $X_{213c}$, $X_{213d}$, $X_{213e}$, $X_{213f}$, $X_{213g}$, $X_{213h}$, $X_{213i}$, $X_{213j}$, $X_{213k}$, $X_{213l}$
Curves that minimally cover $X_{213}$ and have infinitely many rational points. $X_{213a}$, $X_{213b}$, $X_{213c}$, $X_{213d}$, $X_{213e}$, $X_{213f}$, $X_{213g}$, $X_{213h}$, $X_{213i}$, $X_{213j}$, $X_{213k}$, $X_{213l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{213}) = \mathbb{Q}(f_{213}), f_{102} = f_{213}^{2} + 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 1021230x + 397475626$, with conductor $1530$
Generic density of odd order reductions $25/224$