## The modular curve $X_{62i}$

Curve name $X_{62i}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{62}$
Curves that $X_{62i}$ minimally covers
Curves that minimally cover $X_{62i}$
Curves that minimally cover $X_{62i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} + 108t^{10} - 1620t^{8} + 6048t^{6} - 6480t^{4} + 1728t^{2} - 1728$ $B(t) = -54t^{18} + 324t^{16} + 6480t^{14} - 42336t^{12} + 114048t^{10} - 228096t^{8} + 338688t^{6} - 207360t^{4} - 41472t^{2} + 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 181512x + 9647100$, with conductor $12936$
Generic density of odd order reductions $691/5376$