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

Curve name $X_{27b}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right]$
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 $X_{27b}$ minimally covers
Curves that minimally cover $X_{27b}$
Curves that minimally cover $X_{27b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -6912t^{10} + 20736t^{8} + 11232t^{6} + 1296t^{4} - 27t^{2}$ $B(t) = 221184t^{15} + 1990656t^{13} + 953856t^{11} - 59616t^{7} - 7776t^{5} - 54t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 3300x - 776000$, with conductor $7200$
Generic density of odd order reductions $37/224$