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

Curve name $X_{217a}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{75g}$
Meaning/Special name
Chosen covering $X_{217}$
Curves that $X_{217a}$ minimally covers
Curves that minimally cover $X_{217a}$
Curves that minimally cover $X_{217a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} + 53568t^{14} - 1560384t^{12} + 13886208t^{10} - 44357760t^{8} + 55544832t^{6} - 24966144t^{4} + 3428352t^{2} - 27648$ $B(t) = 432t^{24} + 425088t^{22} - 37376640t^{20} + 928295424t^{18} - 10162195200t^{16} + 54276231168t^{14} - 149378052096t^{12} + 217104924672t^{10} - 162595123200t^{8} + 59410907136t^{6} - 9568419840t^{4} + 435290112t^{2} + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 24703x + 579807$, with conductor $1344$
Generic density of odd order reductions $271/2688$