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

Curve name $X_{62g}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
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 $X_{62g}$ minimally covers
Curves that minimally cover $X_{62g}$
Curves that minimally cover $X_{62g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 1296t^{12} + 11232t^{8} - 20736t^{4} - 6912$ $B(t) = 54t^{24} - 7776t^{20} + 59616t^{16} - 953856t^{8} + 1990656t^{4} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 21962992x - 12752438180$, with conductor $142296$
Generic density of odd order reductions $307/2688$