## The modular curve $X_{66}$

Curve name $X_{66}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{24}$
Meaning/Special name
Chosen covering $X_{24}$
Curves that $X_{66}$ minimally covers $X_{24}$, $X_{38}$, $X_{39}$
Curves that minimally cover $X_{66}$ $X_{214}$, $X_{254}$, $X_{256}$, $X_{395}$, $X_{398}$, $X_{66a}$, $X_{66b}$, $X_{66c}$, $X_{66d}$, $X_{66e}$, $X_{66f}$
Curves that minimally cover $X_{66}$ and have infinitely many rational points. $X_{214}$, $X_{66a}$, $X_{66b}$, $X_{66c}$, $X_{66d}$, $X_{66e}$, $X_{66f}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{66}) = \mathbb{Q}(f_{66}), f_{24} = \frac{f_{66}^{2} - 8}{f_{66}^{2} + 8f_{66} + 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 7353x - 242352$, with conductor $10080$
Generic density of odd order reductions $289/1792$