## The modular curve $X_{41}$

Curve name $X_{41}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{11}$
Curves that $X_{41}$ minimally covers $X_{11}$, $X_{15}$, $X_{17}$
Curves that minimally cover $X_{41}$ $X_{80}$, $X_{81}$, $X_{82}$, $X_{131}$, $X_{138}$
Curves that minimally cover $X_{41}$ and have infinitely many rational points. $X_{80}$, $X_{81}$, $X_{82}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{41}) = \mathbb{Q}(f_{41}), f_{11} = \frac{8f_{41}^{2} + 16}{f_{41}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 1591x - 18317$, with conductor $17664$
Generic density of odd order reductions $2659/10752$