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

Curve name $X_{208d}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{96q}$
Meaning/Special name
Chosen covering $X_{208}$
Curves that $X_{208d}$ minimally covers
Curves that minimally cover $X_{208d}$
Curves that minimally cover $X_{208d}$ 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 $y^2 + xy = x^3 - 393046x + 94807076$, with conductor $8670$
Generic density of odd order reductions $271/2688$