## The modular curve $X_{27e}$

Curve name $X_{27e}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{27}$
Meaning/Special name
Chosen covering $X_{27}$
Curves that $X_{27e}$ minimally covers
Curves that minimally cover $X_{27e}$
Curves that minimally cover $X_{27e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1728t^{6} + 6048t^{4} - 108t^{2}$ $B(t) = 27648t^{9} + 228096t^{7} - 57024t^{5} - 432t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 2458x + 18241$, with conductor $1025$
Generic density of odd order reductions $289/1792$