Number field 24.0.572565594852444156646728515625.1: \(\Q(\zeta_{45})\)

• Label: 24.0.572...625.1
• Degree: $24$
• Signature: $[0, 12]$
• Discriminant: $5.726\times 10^{29}$
• Root discriminant: $17.37$
• Ramified primes: $3, 5$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_2\times C_{12}$ (as 24T2)

Normalized defining polynomial

\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \)

Invariants

Degree:		$24$
Signature:		$[0, 12]$
Discriminant:		\(572565594852444156646728515625\)\(\medspace = 3^{36}\cdot 5^{18}\)
Root discriminant:		$17.37$
Ramified primes:		$3, 5$
$|\Gal(K/\Q)|$:		$24$
This field is Galois and abelian over $\Q$.
Conductor:		\(45=3^{2}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{45}(1,·)$, $\chi_{45}(2,·)$, $\chi_{45}(4,·)$, $\chi_{45}(7,·)$, $\chi_{45}(8,·)$, $\chi_{45}(11,·)$, $\chi_{45}(13,·)$, $\chi_{45}(14,·)$, $\chi_{45}(16,·)$, $\chi_{45}(17,·)$, $\chi_{45}(19,·)$, $\chi_{45}(22,·)$, $\chi_{45}(23,·)$, $\chi_{45}(26,·)$, $\chi_{45}(28,·)$, $\chi_{45}(29,·)$, $\chi_{45}(31,·)$, $\chi_{45}(32,·)$, $\chi_{45}(34,·)$, $\chi_{45}(37,·)$, $\chi_{45}(38,·)$, $\chi_{45}(41,·)$, $\chi_{45}(43,·)$, $\chi_{45}(44,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$11$
Torsion generator:		\( -a \) (order $90$)
Fundamental units:		\( a^{9} + 1 \),  \( a^{6} - 1 \),  \( a^{12} - 1 \),  \( a^{22} - a^{17} - a^{2} \),  \( a^{20} - a^{15} + a^{5} \),  \( a^{2} - 1 \),  \( a - 1 \),  \( a^{22} + a^{20} - a^{16} + a^{13} + a^{12} - a^{10} + a^{7} + a^{4} - a \),  \( a^{4} - 1 \),  \( a^{21} - a^{18} - a^{14} - a^{9} + a^{6} - 1 \),  \( a^{22} - a^{18} - a^{16} + a^{13} + a^{4} - a \) (assuming GRH)
Regulator:		\( 2124832.236129185 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 2124832.236129185 \cdot 1}{90\sqrt{572565594852444156646728515625}}\approx 0.118121222580884$ (assuming GRH)

Galois group

$C_2\times C_{12}$ (as 24T2):

An abelian group of order 24

The 24 conjugacy class representatives for $C_2\times C_{12}$

Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 6.0.2460375.1, 6.6.820125.1, \(\Q(\zeta_{9})\), \(\Q(\zeta_{15})\), 12.0.6053445140625.1, \(\Q(\zeta_{45})^+\), 12.0.84075626953125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$	R	R	${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$	${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$	${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$	${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
3	Data not computed
5	Data not computed