## The modular curve $X_{98}$

Curve name $X_{98}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
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 $X_{98}$ minimally covers $X_{25}$
Curves that minimally cover $X_{98}$ $X_{181}$, $X_{187}$, $X_{188}$, $X_{189}$, $X_{193}$, $X_{194}$, $X_{200}$, $X_{204}$, $X_{244}$, $X_{269}$, $X_{272}$, $X_{279}$, $X_{98a}$, $X_{98b}$, $X_{98c}$, $X_{98d}$, $X_{98e}$, $X_{98f}$, $X_{98g}$, $X_{98h}$, $X_{98i}$, $X_{98j}$, $X_{98k}$, $X_{98l}$, $X_{98m}$, $X_{98n}$, $X_{98o}$, $X_{98p}$
Curves that minimally cover $X_{98}$ and have infinitely many rational points. $X_{181}$, $X_{187}$, $X_{188}$, $X_{189}$, $X_{193}$, $X_{194}$, $X_{200}$, $X_{204}$, $X_{98a}$, $X_{98b}$, $X_{98c}$, $X_{98d}$, $X_{98e}$, $X_{98f}$, $X_{98g}$, $X_{98h}$, $X_{98i}$, $X_{98j}$, $X_{98k}$, $X_{98l}$, $X_{98m}$, $X_{98n}$, $X_{98o}$, $X_{98p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{98}) = \mathbb{Q}(f_{98}), f_{25} = f_{98}^{2} - 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 100x - 125$, with conductor $525$
Generic density of odd order reductions $19/168$