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

Curve name $X_{58d}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 4 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$
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 $X_{58d}$ minimally covers
Curves that minimally cover $X_{58d}$
Curves that minimally cover $X_{58d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{14} + 216t^{12} - 1620t^{10} + 3024t^{8} - 1620t^{6} + 216t^{4} - 108t^{2}$ $B(t) = 432t^{21} - 1296t^{19} - 12960t^{17} + 42336t^{15} - 57024t^{13} + 57024t^{11} - 42336t^{9} + 12960t^{7} + 1296t^{5} - 432t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 117005x + 14142372$, with conductor $2925$
Generic density of odd order reductions $307/2688$