Global number field 30.0.17761887753093897979823770061456102763834271.1: \(\Q(\zeta_{31})\)

• Label: 30.0.17761887753...4271.1
• Degree: $30$
• Signature: $[0, 15]$
• Discriminant: $-\,31^{29}$
• Root discriminant: $27.65$
• Ramified prime: $31$
• Class number: $9$ (GRH)
• Class group: $[9]$ (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

\( x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)

Invariants

Degree:		$30$
Signature:		$[0, 15]$
Discriminant:		\(-17761887753093897979823770061456102763834271=-\,31^{29}\)
Root discriminant:		$27.65$
Ramified primes:		$31$
This field is Galois and abelian over $\Q$.
Conductor:		\(31\)
Dirichlet character group:		$\lbrace$$\chi_{31}(1,·)$, $\chi_{31}(2,·)$, $\chi_{31}(3,·)$, $\chi_{31}(4,·)$, $\chi_{31}(5,·)$, $\chi_{31}(6,·)$, $\chi_{31}(7,·)$, $\chi_{31}(8,·)$, $\chi_{31}(9,·)$, $\chi_{31}(10,·)$, $\chi_{31}(11,·)$, $\chi_{31}(12,·)$, $\chi_{31}(13,·)$, $\chi_{31}(14,·)$, $\chi_{31}(15,·)$, $\chi_{31}(16,·)$, $\chi_{31}(17,·)$, $\chi_{31}(18,·)$, $\chi_{31}(19,·)$, $\chi_{31}(20,·)$, $\chi_{31}(21,·)$, $\chi_{31}(22,·)$, $\chi_{31}(23,·)$, $\chi_{31}(24,·)$, $\chi_{31}(25,·)$, $\chi_{31}(26,·)$, $\chi_{31}(27,·)$, $\chi_{31}(28,·)$, $\chi_{31}(29,·)$, $\chi_{31}(30,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$

Class group and class number

$C_{9}$, which has order $9$ (assuming GRH)

Unit group

Rank:		$14$
Torsion generator:		\( a \) (order $62$)
Fundamental units:		\( a - 1 \),  \( a^{2} + 1 \),  \( a^{3} - 1 \),  \( a^{2} - a + 1 \),  \( a^{4} + 1 \),  \( a^{27} + a^{23} - a^{18} - a^{14} + a^{9} + a^{5} - 1 \),  \( a^{22} - a^{11} + 1 \),  \( a^{21} + a^{11} + a \),  \( a^{6} + 1 \),  \( a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1 \),  \( a^{25} + a^{19} + a^{13} \),  \( a^{4} - a^{3} + a^{2} - a + 1 \),  \( a^{17} + a^{3} \),  \( a^{16} - a \) (assuming GRH)
Regulator:		\( 4316173757.895952 \) (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

A cyclic group of order 30

The 30 conjugacy class representatives for $C_{30}$

Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-31}) \), 3.3.961.1, 5.5.923521.1, 6.0.28629151.1, 10.0.26439622160671.1, \(\Q(\zeta_{31})^+\)

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	${\href{/LocalNumberField/2.5.0.1}{5} }^{6}$	$30$	${\href{/LocalNumberField/5.3.0.1}{3} }^{10}$	$15^{2}$	$30$	$30$	$30$	$15^{2}$	${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$	R	${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$	$15^{2}$	$30$	${\href{/LocalNumberField/47.5.0.1}{5} }^{6}$	$30$	$15^{2}$

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
31	Data not computed