## The modular curve $X_{208f}$

Curve name $X_{208f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{96e}$
Meaning/Special name
Chosen covering $X_{208}$
Curves that $X_{208f}$ minimally covers
Curves that minimally cover $X_{208f}$
Curves that minimally cover $X_{208f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{24} + 54t^{20} - 54t^{12} + 54t^{4} - 27$ $B(t) = 54t^{36} - 162t^{32} + 81t^{28} + 189t^{24} - 324t^{20} + 324t^{16} - 189t^{12} - 81t^{8} + 162t^{4} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 22416751x - 40853279602$, with conductor $6150$
Generic density of odd order reductions $73/672$