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

Curve name $X_{44a}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right]$
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 $X_{44a}$ minimally covers
Curves that minimally cover $X_{44a}$
Curves that minimally cover $X_{44a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} - 2592t^{6} + 11232t^{4} - 10368t^{2} - 1728$ $B(t) = 432t^{12} - 31104t^{10} + 119232t^{8} - 476928t^{4} + 497664t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 11564x + 378672$, with conductor $3136$
Generic density of odd order reductions $289/1792$