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
LevelIndex of imageCorresponding 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 \[\mathbb{P}^{1}, \mathbb{Q}(X_{30}) = \mathbb{Q}(f_{30}), f_{12} = -f_{30}^{2}\]
Info about rational points None
Comments on finding rational points None
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 $419/1344$

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