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

Curve name $X_{58e}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
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 $X_{58e}$ minimally covers
Curves that minimally cover $X_{58e}$
Curves that minimally cover $X_{58e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} - 1512t^{4} - 108$ $B(t) = 432t^{12} - 14256t^{8} - 14256t^{4} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 33281x - 2129919$, with conductor $12480$
Generic density of odd order reductions $307/2688$