Global number field 24.24.9606056659007943744000000000000000000.1: \(\Q(\zeta_{180})^+\)

• Label: 24.24.9606056659...0000.1
• Degree: $24$
• Signature: $[24, 0]$
• Discriminant: $2^{24}\cdot 3^{36}\cdot 5^{18}$
• Root discriminant: $34.75$
• Ramified primes: $2, 3, 5$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{12}$ (as 24T2)

Normalized defining polynomial

\( x^{24} - 24 x^{22} + 252 x^{20} - 1519 x^{18} + 5796 x^{16} - 14553 x^{14} + 24206 x^{12} - 26169 x^{10} + 17523 x^{8} - 6623 x^{6} + 1182 x^{4} - 72 x^{2} + 1 \)

Invariants

Degree:		$24$
Signature:		$[24, 0]$
Discriminant:		\(9606056659007943744000000000000000000=2^{24}\cdot 3^{36}\cdot 5^{18}\)
Root discriminant:		$34.75$
Ramified primes:		$2, 3, 5$
This field is Galois and abelian over $\Q$.
Conductor:		\(180=2^{2}\cdot 3^{2}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(131,·)$, $\chi_{180}(7,·)$, $\chi_{180}(137,·)$, $\chi_{180}(11,·)$, $\chi_{180}(77,·)$, $\chi_{180}(109,·)$, $\chi_{180}(17,·)$, $\chi_{180}(67,·)$, $\chi_{180}(71,·)$, $\chi_{180}(163,·)$, $\chi_{180}(113,·)$, $\chi_{180}(103,·)$, $\chi_{180}(169,·)$, $\chi_{180}(43,·)$, $\chi_{180}(173,·)$, $\chi_{180}(49,·)$, $\chi_{180}(179,·)$, $\chi_{180}(53,·)$, $\chi_{180}(119,·)$, $\chi_{180}(121,·)$, $\chi_{180}(59,·)$, $\chi_{180}(61,·)$, $\chi_{180}(127,·)$$\rbrace$
This is not 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:		$23$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 46780365797.01783 \) (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

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), 6.6.157464000.1, 6.6.820125.1, \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{60})^+\), 12.12.24794911296000000.1, 12.12.344373768000000000.1, \(\Q(\zeta_{45})^+\)

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	R	R	R	${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$	${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$	${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$	${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{8}$

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
2	Data not computed
3	Data not computed
5	Data not computed