## The modular curve $X_{61}$

Curve name $X_{61}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{24}$
Meaning/Special name
Chosen covering $X_{24}$
Curves that $X_{61}$ minimally covers $X_{24}$, $X_{35}$, $X_{45}$
Curves that minimally cover $X_{61}$ $X_{209}$, $X_{210}$, $X_{251}$, $X_{254}$, $X_{392}$, $X_{393}$, $X_{61a}$, $X_{61b}$, $X_{61c}$, $X_{61d}$, $X_{61e}$, $X_{61f}$, $X_{61g}$, $X_{61h}$
Curves that minimally cover $X_{61}$ and have infinitely many rational points. $X_{209}$, $X_{210}$, $X_{61a}$, $X_{61b}$, $X_{61c}$, $X_{61d}$, $X_{61e}$, $X_{61f}$, $X_{61g}$, $X_{61h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{61}) = \mathbb{Q}(f_{61}), f_{24} = f_{61}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 4226x + 94223$, with conductor $7275$
Generic density of odd order reductions $289/1792$