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

Curve name $X_{217h}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{75h}$
Meaning/Special name
Chosen covering $X_{217}$
Curves that $X_{217h}$ minimally covers
Curves that minimally cover $X_{217h}$
Curves that minimally cover $X_{217h}$ 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 $y^2 + xy = x^3 + 18913x - 381333$, with conductor $294$
Generic density of odd order reductions $81/896$