## The modular curve $X_{77}$

Curve name $X_{77}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{29}$
Curves that $X_{77}$ minimally covers $X_{29}$, $X_{43}$, $X_{45}$
Curves that minimally cover $X_{77}$ $X_{216}$, $X_{257}$, $X_{258}$, $X_{275}$, $X_{276}$, $X_{373}$, $X_{391}$
Curves that minimally cover $X_{77}$ and have infinitely many rational points. $X_{216}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{77}) = \mathbb{Q}(f_{77}), f_{29} = \frac{f_{77}^{2} - \frac{1}{2}}{f_{77}}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 307x - 4587$, with conductor $5376$
Generic density of odd order reductions $401/1792$