## The modular curve $X_{101}$

Curve name $X_{101}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{25}$
Curves that $X_{101}$ minimally covers $X_{25}$
Curves that minimally cover $X_{101}$ $X_{183}$, $X_{185}$, $X_{188}$, $X_{189}$, $X_{190}$, $X_{192}$, $X_{200}$, $X_{203}$, $X_{245}$, $X_{268}$, $X_{274}$, $X_{279}$, $X_{101a}$, $X_{101b}$, $X_{101c}$, $X_{101d}$, $X_{101e}$, $X_{101f}$, $X_{101g}$, $X_{101h}$, $X_{101i}$, $X_{101j}$, $X_{101k}$, $X_{101l}$, $X_{101m}$, $X_{101n}$, $X_{101o}$, $X_{101p}$
Curves that minimally cover $X_{101}$ and have infinitely many rational points. $X_{183}$, $X_{185}$, $X_{188}$, $X_{189}$, $X_{190}$, $X_{192}$, $X_{200}$, $X_{203}$, $X_{101a}$, $X_{101b}$, $X_{101c}$, $X_{101d}$, $X_{101e}$, $X_{101f}$, $X_{101g}$, $X_{101h}$, $X_{101i}$, $X_{101j}$, $X_{101k}$, $X_{101l}$, $X_{101m}$, $X_{101n}$, $X_{101o}$, $X_{101p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{101}) = \mathbb{Q}(f_{101}), f_{25} = 8f_{101}^{2} - 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 1225x - 17000$, with conductor $525$
Generic density of odd order reductions $19/168$