## The modular curve $X_{33}$

Curve name $X_{33}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{13}$
Curves that $X_{33}$ minimally covers $X_{13}$
Curves that minimally cover $X_{33}$ $X_{78}$, $X_{100}$, $X_{33a}$, $X_{33b}$, $X_{33c}$, $X_{33d}$, $X_{33e}$, $X_{33f}$, $X_{33g}$, $X_{33h}$
Curves that minimally cover $X_{33}$ and have infinitely many rational points. $X_{78}$, $X_{100}$, $X_{33a}$, $X_{33b}$, $X_{33c}$, $X_{33d}$, $X_{33e}$, $X_{33f}$, $X_{33g}$, $X_{33h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{33}) = \mathbb{Q}(f_{33}), f_{13} = -2f_{33}^{2} - 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 18x + 4$, with conductor $198$
Generic density of odd order reductions $513/3584$