## The modular curve $X_{181}$

Curve name $X_{181}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{87}$
Curves that $X_{181}$ minimally covers $X_{87}$, $X_{98}$, $X_{100}$
Curves that minimally cover $X_{181}$ $X_{448}$, $X_{450}$, $X_{455}$, $X_{459}$, $X_{181a}$, $X_{181b}$, $X_{181c}$, $X_{181d}$, $X_{181e}$, $X_{181f}$, $X_{181g}$, $X_{181h}$
Curves that minimally cover $X_{181}$ and have infinitely many rational points. $X_{181a}$, $X_{181b}$, $X_{181c}$, $X_{181d}$, $X_{181e}$, $X_{181f}$, $X_{181g}$, $X_{181h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{181}) = \mathbb{Q}(f_{181}), f_{87} = \frac{f_{181}^{2} - \frac{1}{8}}{f_{181}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 1026x + 10692$, with conductor $306$
Generic density of odd order reductions $635/5376$