## The modular curve $X_{217b}$

Curve name $X_{217b}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{75j}$
Meaning/Special name
Chosen covering $X_{217}$
Curves that $X_{217b}$ minimally covers
Curves that minimally cover $X_{217b}$
Curves that minimally cover $X_{217b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} + 13392t^{14} - 390096t^{12} + 3471552t^{10} - 11089440t^{8} + 13886208t^{6} - 6241536t^{4} + 857088t^{2} - 6912$ $B(t) = -54t^{24} - 53136t^{22} + 4672080t^{20} - 116036928t^{18} + 1270274400t^{16} - 6784528896t^{14} + 18672256512t^{12} - 27138115584t^{10} + 20324390400t^{8} - 7426363392t^{6} + 1196052480t^{4} - 54411264t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 6176x - 69388$, with conductor $336$
Generic density of odd order reductions $53/896$