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

Curve name $X_{217f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
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 $X_{217f}$ minimally covers
Curves that minimally cover $X_{217f}$
Curves that minimally cover $X_{217f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{24} + 14040t^{22} - 715608t^{20} + 14872032t^{18} - 154987344t^{16} + 845372160t^{14} - 2364615936t^{12} + 3381488640t^{10} - 2479797504t^{8} + 951810048t^{6} - 183195648t^{4} + 14376960t^{2} - 110592$ $B(t) = -54t^{36} - 51192t^{34} + 6561000t^{32} - 307715328t^{30} + 7629033600t^{28} - 113548055040t^{26} + 1069175794176t^{24} - 6481319473152t^{22} + 25288532315136t^{20} - 63220124602368t^{18} + 101154129260544t^{16} - 103701111570432t^{14} + 68427250827264t^{12} - 29068302090240t^{10} + 7812130406400t^{8} - 1260401983488t^{6} + 107495424000t^{4} - 3354918912t^{2} - 14155776$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 302608x + 24405312$, with conductor $2352$
Generic density of odd order reductions $299/2688$