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

Curve name $X_{58k}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{58k}$ minimally covers
Curves that minimally cover $X_{58k}$
Curves that minimally cover $X_{58k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{10} - 378t^{6} - 27t^{2}$ $B(t) = 54t^{15} - 1782t^{11} - 1782t^{7} + 54t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 13000x - 528125$, with conductor $975$
Generic density of odd order reductions $25/224$