## The modular curve $X_{44b}$

Curve name $X_{44b}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{44}$
Curves that $X_{44b}$ minimally covers
Curves that minimally cover $X_{44b}$
Curves that minimally cover $X_{44b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} - 648t^{6} + 2808t^{4} - 2592t^{2} - 432$ $B(t) = 54t^{12} - 3888t^{10} + 14904t^{8} - 59616t^{4} + 62208t^{2} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 2891x + 47334$, with conductor $392$
Generic density of odd order reductions $643/5376$