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

Curve name $X_{217e}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{75e}$
Meaning/Special name
Chosen covering $X_{217}$
Curves that $X_{217e}$ minimally covers
Curves that minimally cover $X_{217e}$
Curves that minimally cover $X_{217e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{24} + 56160t^{22} - 2862432t^{20} + 59488128t^{18} - 619949376t^{16} + 3381488640t^{14} - 9458463744t^{12} + 13525954560t^{10} - 9919190016t^{8} + 3807240192t^{6} - 732782592t^{4} + 57507840t^{2} - 442368$ $B(t) = 432t^{36} + 409536t^{34} - 52488000t^{32} + 2461722624t^{30} - 61032268800t^{28} + 908384440320t^{26} - 8553406353408t^{24} + 51850555785216t^{22} - 202308258521088t^{20} + 505760996818944t^{18} - 809233034084352t^{16} + 829608892563456t^{14} - 547418006618112t^{12} + 232546416721920t^{10} - 62497043251200t^{8} + 10083215867904t^{6} - 859963392000t^{4} + 26839351296t^{2} + 113246208$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 1210431x - 196452927$, with conductor $9408$
Generic density of odd order reductions $109/896$