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

Curve name $X_{33e}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right]$
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 $X_{33e}$ minimally covers
Curves that minimally cover $X_{33e}$
Curves that minimally cover $X_{33e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{4} - 216t^{2} - 108$ $B(t) = 54t^{6} + 648t^{4} + 1620t^{2} - 432$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 2x - 1$, with conductor $66$
Generic density of odd order reductions $643/5376$