## The modular curve $X_{225k}$

Curve name $X_{225k}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{85p}$
Meaning/Special name
Chosen covering $X_{225}$
Curves that $X_{225k}$ minimally covers
Curves that minimally cover $X_{225k}$
Curves that minimally cover $X_{225k}$ 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 $y^2 = x^3 - x^2 - 864008x + 309406512$, with conductor $1200$
Generic density of odd order reductions $5/42$