## The modular curve $X_{227k}$

Curve name $X_{227k}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $48$ $X_{84l}$
Meaning/Special name
Chosen covering $X_{227}$
Curves that $X_{227k}$ minimally covers
Curves that minimally cover $X_{227k}$
Curves that minimally cover $X_{227k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1769472t^{16} + 53084160t^{14} - 59719680t^{12} + 23224320t^{10} - 7561728t^{8} + 1451520t^{6} - 233280t^{4} + 12960t^{2} - 27$ $B(t) = 905969664t^{24} + 57076088832t^{22} - 235438866432t^{20} + 217602588672t^{18} - 117836218368t^{16} + 43252973568t^{14} - 12875563008t^{12} + 2703310848t^{10} - 460297728t^{8} + 53125632t^{6} - 3592512t^{4} + 54432t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 + 6550x - 962215$, with conductor $510$
Generic density of odd order reductions $19/336$