## The modular curve $X_{212}$

Curve name $X_{212}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{86}$
Meaning/Special name
Chosen covering $X_{86}$
Curves that $X_{212}$ minimally covers $X_{86}$, $X_{118}$, $X_{122}$
Curves that minimally cover $X_{212}$ $X_{478}$, $X_{481}$, $X_{493}$, $X_{498}$, $X_{212a}$, $X_{212b}$, $X_{212c}$, $X_{212d}$, $X_{212e}$, $X_{212f}$, $X_{212g}$, $X_{212h}$, $X_{212i}$, $X_{212j}$, $X_{212k}$, $X_{212l}$
Curves that minimally cover $X_{212}$ and have infinitely many rational points. $X_{212a}$, $X_{212b}$, $X_{212c}$, $X_{212d}$, $X_{212e}$, $X_{212f}$, $X_{212g}$, $X_{212h}$, $X_{212i}$, $X_{212j}$, $X_{212k}$, $X_{212l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{212}) = \mathbb{Q}(f_{212}), f_{86} = \frac{f_{212}}{f_{212}^{2} - \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 5391x + 285606$, with conductor $735$
Generic density of odd order reductions $25/224$