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 


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 \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 774144t^{8}  2211840t^{7}  2598912t^{6}  1714176t^{5}  736128t^{4}  214272t^{3}  40608t^{2}  4320t  189\] \[B(t) = 240648192t^{12} + 976748544t^{11} + 1677459456t^{10} + 1583677440t^{9} + 875225088t^{8} + 253476864t^{7}  31684608t^{5}  13675392t^{4}  3093120t^{3}  409536t^{2}  29808t  918\]  
Info about rational points  
Comments on finding rational points  None  
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  $1427/5376$ 