## The modular curve $X_{58}$

Curve name $X_{58}$
Index $24$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$
Meaning/Special name
Chosen covering $X_{23}$
Curves that $X_{58}$ minimally covers $X_{23}$, $X_{24}$, $X_{25}$
Curves that minimally cover $X_{58}$ $X_{183}$, $X_{186}$, $X_{187}$, $X_{189}$, $X_{248}$, $X_{249}$, $X_{251}$, $X_{255}$, $X_{357}$, $X_{58a}$, $X_{58b}$, $X_{58c}$, $X_{58d}$, $X_{58e}$, $X_{58f}$, $X_{58g}$, $X_{58h}$, $X_{58i}$, $X_{58j}$, $X_{58k}$
Curves that minimally cover $X_{58}$ and have infinitely many rational points. $X_{183}$, $X_{187}$, $X_{189}$, $X_{58a}$, $X_{58b}$, $X_{58c}$, $X_{58d}$, $X_{58e}$, $X_{58f}$, $X_{58g}$, $X_{58h}$, $X_{58i}$, $X_{58j}$, $X_{58k}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{58}) = \mathbb{Q}(f_{58}), f_{23} = \frac{f_{58}^{2} + 2f_{58} - 1}{f_{58}^{2} + 1}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 40807260x - 94524898325$, with conductor $6435$
Generic density of odd order reductions $19/168$