## The modular curve $X_{189l}$

Curve name $X_{189l}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{58}$
Meaning/Special name
Chosen covering $X_{189}$
Curves that $X_{189l}$ minimally covers
Curves that minimally cover $X_{189l}$
Curves that minimally cover $X_{189l}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 3456t^{15} - 48384t^{14} - 387072t^{13} - 2322432t^{12} - 15482880t^{11} - 108380160t^{10} - 571539456t^{9} - 2002157568t^{8} - 4572315648t^{7} - 6936330240t^{6} - 7927234560t^{5} - 9512681472t^{4} - 12683575296t^{3} - 12683575296t^{2} - 7247757312t - 1811939328$ $B(t) = 432t^{24} + 20736t^{23} + 456192t^{22} + 6082560t^{21} + 51093504t^{20} + 204374016t^{19} - 1021870080t^{18} - 23473815552t^{17} - 199994572800t^{16} - 1105566105600t^{15} - 4510709710848t^{14} - 14896859185152t^{13} - 43694916894720t^{12} - 119174873481216t^{11} - 288685421494272t^{10} - 566049846067200t^{9} - 819177770188800t^{8} - 769189988007936t^{7} - 267877110251520t^{6} + 428603376402432t^{5} + 857206752804864t^{4} + 816387383623680t^{3} + 489832430174208t^{2} + 178120883699712t + 29686813949952$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 483201x - 126236799$, with conductor $6720$
Generic density of odd order reductions $299/2688$