## The modular curve $X_{30}$

Curve name $X_{30}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{12}$
Meaning/Special name
Chosen covering $X_{12}$
Curves that $X_{30}$ minimally covers $X_{12}$
Curves that minimally cover $X_{30}$ $X_{72}$, $X_{88}$, $X_{89}$, $X_{93}$, $X_{103}$, $X_{104}$, $X_{126}$, $X_{142}$, $X_{162}$, $X_{163}$, $X_{164}$, $X_{167}$, $X_{172}$, $X_{174}$
Curves that minimally cover $X_{30}$ and have infinitely many rational points. $X_{89}$, $X_{93}$, $X_{103}$, $X_{104}$, $X_{167}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{30}) = \mathbb{Q}(f_{30}), f_{12} = -f_{30}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 7325x + 219000$, with conductor $63075$
Generic density of odd order reductions $419/1344$