## The modular curve $X_{181g}$

Curve name $X_{181g}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{181}$
Curves that $X_{181g}$ minimally covers
Curves that minimally cover $X_{181g}$
Curves that minimally cover $X_{181g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1811939328t^{16} - 1811939328t^{14} - 339738624t^{12} + 28311552t^{10} - 101744640t^{8} + 442368t^{6} - 82944t^{4} - 6912t^{2} - 108$ $B(t) = -29686813949952t^{24} - 44530220924928t^{22} - 19481971654656t^{20} - 1623497637888t^{18} + 4065991852032t^{16} + 1891664658432t^{14} - 117323071488t^{12} + 29557260288t^{10} + 992673792t^{8} - 6193152t^{6} - 1161216t^{4} - 41472t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 7297x + 195455$, with conductor $3264$
Generic density of odd order reductions $271/2688$