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

Curve name $X_{217d}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $48$ $X_{75f}$
Meaning/Special name
Chosen covering $X_{217}$
Curves that $X_{217d}$ minimally covers
Curves that minimally cover $X_{217d}$
Curves that minimally cover $X_{217d}$ 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 $271/2688$