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

Curve name $X_{208k}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{96c}$
Meaning/Special name
Chosen covering $X_{208}$
Curves that $X_{208k}$ minimally covers
Curves that minimally cover $X_{208k}$
Curves that minimally cover $X_{208k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{24} + 216t^{20} - 216t^{12} + 216t^{4} - 108$ $B(t) = 432t^{36} - 1296t^{32} + 648t^{28} + 1512t^{24} - 2592t^{20} + 2592t^{16} - 1512t^{12} - 648t^{8} + 1296t^{4} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 1434672033x + 20915444484063$, with conductor $196800$
Generic density of odd order reductions $299/2688$