## The modular curve $X_{212g}$

Curve name $X_{212g}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{86o}$
Meaning/Special name
Chosen covering $X_{212}$
Curves that $X_{212g}$ minimally covers
Curves that minimally cover $X_{212g}$
Curves that minimally cover $X_{212g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1855425871872t^{24} + 27831388078080t^{22} - 15713137852416t^{20} + 3913788948480t^{18} - 985242009600t^{16} + 149484994560t^{14} - 23130537984t^{12} + 2335703040t^{10} - 240537600t^{8} + 14929920t^{6} - 936576t^{4} + 25920t^{2} - 27$ $B(t) = 972777519512027136t^{36} + 30642491864628854784t^{34} - 63154540524569886720t^{32} + 30642491864628854784t^{30} - 10869648806891225088t^{28} + 2842809303847403520t^{26} - 632982247040876544t^{24} + 112216156730818560t^{22} - 18089358573699072t^{20} + 2348795207614464t^{18} - 282646227714048t^{16} + 27396522639360t^{14} - 2414635646976t^{12} + 169444638720t^{10} - 10123149312t^{8} + 445906944t^{6} - 14359680t^{4} + 108864t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 2751x - 104477$, with conductor $75$
Generic density of odd order reductions $11/112$