## The modular curve $X_{58a}$

Curve name $X_{58a}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{58}$
Meaning/Special name
Chosen covering $X_{58}$
Curves that $X_{58a}$ minimally covers
Curves that minimally cover $X_{58a}$
Curves that minimally cover $X_{58a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 54t^{10} - 405t^{8} - 756t^{6} - 405t^{4} - 54t^{2} - 27$ $B(t) = 54t^{18} + 162t^{16} - 1620t^{14} - 5292t^{12} - 7128t^{10} - 7128t^{8} - 5292t^{6} - 1620t^{4} + 162t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 87884x - 9194443$, with conductor $2535$
Generic density of odd order reductions $17/168$