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

Curve name $X_{225f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{85i}$
Meaning/Special name
Chosen covering $X_{225}$
Curves that $X_{225f}$ minimally covers
Curves that minimally cover $X_{225f}$
Curves that minimally cover $X_{225f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1811939328t^{16} - 27179089920t^{14} - 15288238080t^{12} - 2972712960t^{10} - 483950592t^{8} - 46448640t^{6} - 3732480t^{4} - 103680t^{2} - 108$ $B(t) = -29686813949952t^{24} + 935134639423488t^{22} + 1928715193810944t^{20} + 891300203200512t^{18} + 241328575217664t^{16} + 44291044933632t^{14} + 6592288260096t^{12} + 692047577088t^{10} + 58918109184t^{8} + 3400040448t^{6} + 114960384t^{4} + 870912t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 138241x + 19829665$, with conductor $960$
Generic density of odd order reductions $299/2688$