## The modular curve $X_{37b}$

Curve name $X_{37b}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{9}$
Meaning/Special name
Chosen covering $X_{37}$
Curves that $X_{37b}$ minimally covers
Curves that minimally cover $X_{37b}$
Curves that minimally cover $X_{37b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -774144t^{8} - 2211840t^{7} - 2598912t^{6} - 1714176t^{5} - 736128t^{4} - 214272t^{3} - 40608t^{2} - 4320t - 189$ $B(t) = 240648192t^{12} + 976748544t^{11} + 1677459456t^{10} + 1583677440t^{9} + 875225088t^{8} + 253476864t^{7} - 31684608t^{5} - 13675392t^{4} - 3093120t^{3} - 409536t^{2} - 29808t - 918$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 66x + 230$, with conductor $640$
Generic density of odd order reductions $1427/5376$