## The modular curve $X_{67}$

Curve name $X_{67}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 4 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$
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 $X_{67}$ minimally covers $X_{24}$, $X_{28}$, $X_{43}$
Curves that minimally cover $X_{67}$ $X_{255}$, $X_{256}$, $X_{67a}$, $X_{67b}$, $X_{67c}$, $X_{67d}$
Curves that minimally cover $X_{67}$ and have infinitely many rational points. $X_{67a}$, $X_{67b}$, $X_{67c}$, $X_{67d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{67}) = \mathbb{Q}(f_{67}), f_{24} = \frac{-2}{f_{67}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 63x + 162$, with conductor $360$
Generic density of odd order reductions $13/84$