## The modular curve $X_{10c}$

Curve name $X_{10c}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{10}$
Meaning/Special name
Chosen covering $X_{10}$
Curves that $X_{10c}$ minimally covers
Curves that minimally cover $X_{10c}$
Curves that minimally cover $X_{10c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = 324t^{8} + 540t^{6} + 108t^{4} - 108t^{2}$ $B(t) = 3888t^{11} + 12096t^{9} + 12960t^{7} + 5184t^{5} + 432t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 92x + 312$, with conductor $100$
Generic density of odd order reductions $2659/10752$