## The modular curve $X_{100}$

Curve name $X_{100}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
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 $X_{100}$ minimally covers $X_{25}$, $X_{33}$, $X_{34}$
Curves that minimally cover $X_{100}$ $X_{181}$, $X_{182}$, $X_{188}$, $X_{190}$, $X_{247}$, $X_{248}$, $X_{100a}$, $X_{100b}$, $X_{100c}$, $X_{100d}$, $X_{100e}$, $X_{100f}$, $X_{100g}$, $X_{100h}$, $X_{100i}$, $X_{100j}$
Curves that minimally cover $X_{100}$ and have infinitely many rational points. $X_{181}$, $X_{188}$, $X_{190}$, $X_{100a}$, $X_{100b}$, $X_{100c}$, $X_{100d}$, $X_{100e}$, $X_{100f}$, $X_{100g}$, $X_{100h}$, $X_{100i}$, $X_{100j}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{100}) = \mathbb{Q}(f_{100}), f_{25} = \frac{2}{f_{100}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 18759x - 980390$, with conductor $5544$
Generic density of odd order reductions $643/5376$