## The modular curve $X_{44}$

Curve name $X_{44}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{13}$
Curves that $X_{44}$ minimally covers $X_{13}$, $X_{14}$, $X_{17}$
Curves that minimally cover $X_{44}$ $X_{78}$, $X_{79}$, $X_{44a}$, $X_{44b}$, $X_{44c}$, $X_{44d}$
Curves that minimally cover $X_{44}$ and have infinitely many rational points. $X_{78}$, $X_{79}$, $X_{44a}$, $X_{44b}$, $X_{44c}$, $X_{44d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{44}) = \mathbb{Q}(f_{44}), f_{13} = \frac{8f_{44}^{2} + 16}{f_{44}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 6692x - 209034$, with conductor $350$
Generic density of odd order reductions $513/3584$