Global number field 28.0.3053134545970524535745336759489912159909.1: \(\Q(\zeta_{29})\)

• Label: 28.0.30531345459...9909.1
• Degree: $28$
• Signature: $[0, 14]$
• Discriminant: $29^{27}$
• Root discriminant: $25.71$
• Ramified prime: $29$
• Class number: $8$ (GRH)
• Class group: $[2, 2, 2]$ (GRH)
• Galois group: $C_{28}$ (as 28T1)

Normalized defining polynomial

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

Invariants

Degree:		$28$
Signature:		$[0, 14]$
Discriminant:		\(3053134545970524535745336759489912159909=29^{27}\)
Root discriminant:		$25.71$
Ramified primes:		$29$
This field is Galois and abelian over $\Q$.
Conductor:		\(29\)
Dirichlet character group:		$\lbrace$$\chi_{29}(1,·)$, $\chi_{29}(2,·)$, $\chi_{29}(3,·)$, $\chi_{29}(4,·)$, $\chi_{29}(5,·)$, $\chi_{29}(6,·)$, $\chi_{29}(7,·)$, $\chi_{29}(8,·)$, $\chi_{29}(9,·)$, $\chi_{29}(10,·)$, $\chi_{29}(11,·)$, $\chi_{29}(12,·)$, $\chi_{29}(13,·)$, $\chi_{29}(14,·)$, $\chi_{29}(15,·)$, $\chi_{29}(16,·)$, $\chi_{29}(17,·)$, $\chi_{29}(18,·)$, $\chi_{29}(19,·)$, $\chi_{29}(20,·)$, $\chi_{29}(21,·)$, $\chi_{29}(22,·)$, $\chi_{29}(23,·)$, $\chi_{29}(24,·)$, $\chi_{29}(25,·)$, $\chi_{29}(26,·)$, $\chi_{29}(27,·)$, $\chi_{29}(28,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( a \) (order $58$)
Fundamental units:		\( a - 1 \),  \( a^{2} + 1 \),  \( a^{3} - 1 \),  \( a^{2} - a + 1 \),  \( a^{6} + 1 \),  \( a^{4} + 1 \),  \( a^{16} - a^{3} \),  \( a^{4} - a^{3} + a^{2} - a + 1 \),  \( a^{20} + a^{10} + 1 \),  \( a^{20} - a^{11} + a^{2} \),  \( a^{27} + a^{25} + a^{23} + a^{21} + a^{19} + a^{17} + a^{15} + a^{13} + a^{11} \),  \( a^{22} - a^{15} + a^{8} - a \),  \( a^{17} + a^{3} \) (assuming GRH)
Regulator:		\( 487075979.1876791 \) (assuming GRH)

Galois group

$C_{28}$ (as 28T1):

A cyclic group of order 28

The 28 conjugacy class representatives for $C_{28}$

Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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	$28$	$28$	${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$	$28$	${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$	$28$	${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$	R	$28$	$28$	${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$	$28$	$28$	${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$	${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
29	Data not computed