## The modular curve $X_{2b}$

Curve name $X_{2b}$
Index $4$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 3 & 0 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $2$ $X_{2}$ $4$ $2$ $X_{2}$
Meaning/Special name
Chosen covering $X_{2}$
Curves that $X_{2b}$ minimally covers
Curves that minimally cover $X_{2b}$
Curves that minimally cover $X_{2b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{6} - 559872t^{4} - 967458816t^{2} - 557256278016$ $B(t) = 432t^{9} + 2985984t^{7} + 7739670528t^{5} + 8916100448256t^{3} + 3851755393646592t$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 4345x + 64840$, with conductor $1922$
Generic density of odd order reductions $7681/10752$