## The modular curve $X_{18}$

Curve name $X_{18}$
Index $6$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $3$ $X_{6}$
Meaning/Special name Elliptic curves with discriminant $\Delta$ whose $2$-isogenous curve has discriminant in the square class of $-2\Delta$
Chosen covering $X_{6}$
Curves that $X_{18}$ minimally covers $X_{6}$
Curves that minimally cover $X_{18}$ $X_{37}$, $X_{42}$, $X_{46}$, $X_{48}$
Curves that minimally cover $X_{18}$ and have infinitely many rational points. $X_{37}$, $X_{42}$, $X_{46}$, $X_{48}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{18}) = \mathbb{Q}(f_{18}), f_{6} = 8f_{18}^{2} - 16$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 71x + 265$, with conductor $490$
Generic density of odd order reductions $5123/21504$