## The modular curve $X_{33c}$

Curve name $X_{33c}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{33}$
Curves that $X_{33c}$ minimally covers
Curves that minimally cover $X_{33c}$
Curves that minimally cover $X_{33c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} - 648t^{6} - 5292t^{4} - 15552t^{2} - 6912$ $B(t) = 54t^{12} + 1944t^{10} + 27540t^{8} + 190512t^{6} + 632448t^{4} + 746496t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 156x + 736$, with conductor $288$
Generic density of odd order reductions $643/5376$