## The modular curve $X_{211}$

Curve name $X_{211}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 9 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 9 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 5 \end{matrix}\right]$
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 $X_{211}$ minimally covers $X_{85}$, $X_{118}$, $X_{122}$
Curves that minimally cover $X_{211}$ $X_{472}$, $X_{477}$, $X_{478}$, $X_{482}$, $X_{489}$, $X_{490}$, $X_{511}$, $X_{512}$, $X_{513}$, $X_{514}$, $X_{522}$, $X_{528}$, $X_{530}$, $X_{532}$, $X_{211a}$, $X_{211b}$, $X_{211c}$, $X_{211d}$, $X_{211e}$, $X_{211f}$, $X_{211g}$, $X_{211h}$, $X_{211i}$, $X_{211j}$, $X_{211k}$, $X_{211l}$, $X_{211m}$, $X_{211n}$, $X_{211o}$, $X_{211p}$, $X_{211q}$, $X_{211r}$, $X_{211s}$, $X_{211t}$
Curves that minimally cover $X_{211}$ and have infinitely many rational points. $X_{211a}$, $X_{211b}$, $X_{211c}$, $X_{211d}$, $X_{211e}$, $X_{211f}$, $X_{211g}$, $X_{211h}$, $X_{211i}$, $X_{211j}$, $X_{211k}$, $X_{211l}$, $X_{211m}$, $X_{211n}$, $X_{211o}$, $X_{211p}$, $X_{211q}$, $X_{211r}$, $X_{211s}$, $X_{211t}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{211}) = \mathbb{Q}(f_{211}), f_{85} = \frac{2f_{211}^{2} + 8}{f_{211}^{2} + 4f_{211} - 4}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 14435303x + 21587495048$, with conductor $25410$
Generic density of odd order reductions $17/168$