## The modular curve $X_{62}$

Curve name $X_{62}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{25}$
Curves that $X_{62}$ minimally covers $X_{25}$, $X_{38}$, $X_{46}$
Curves that minimally cover $X_{62}$ $X_{198}$, $X_{200}$, $X_{203}$, $X_{204}$, $X_{246}$, $X_{247}$, $X_{62a}$, $X_{62b}$, $X_{62c}$, $X_{62d}$, $X_{62e}$, $X_{62f}$, $X_{62g}$, $X_{62h}$, $X_{62i}$, $X_{62j}$
Curves that minimally cover $X_{62}$ and have infinitely many rational points. $X_{200}$, $X_{203}$, $X_{204}$, $X_{62a}$, $X_{62b}$, $X_{62c}$, $X_{62d}$, $X_{62e}$, $X_{62f}$, $X_{62g}$, $X_{62h}$, $X_{62i}$, $X_{62j}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{62}) = \mathbb{Q}(f_{62}), f_{25} = \frac{f_{62}^{2} + 2}{f_{62}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 33339x + 754630$, with conductor $5544$
Generic density of odd order reductions $643/5376$