## The modular curve $X_{185e}$

Curve name $X_{185e}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{185}$
Curves that $X_{185e}$ minimally covers
Curves that minimally cover $X_{185e}$
Curves that minimally cover $X_{185e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -442368t^{24} - 6856704t^{20} - 7050240t^{16} - 2681856t^{12} - 440640t^{8} - 26784t^{4} - 108$ $B(t) = -113246208t^{36} + 3482320896t^{32} + 10022289408t^{28} + 9314500608t^{24} + 4238770176t^{20} + 1059692544t^{16} + 145539072t^{12} + 9787392t^{8} + 212544t^{4} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 216033x + 38556063$, with conductor $4800$
Generic density of odd order reductions $271/2688$