Number field 28.28.819569564076950716311987772907236898045117333504.1: \(\Q(\zeta_{116})^+\)

• Label: 28.28.819...504.1
• Degree: $28$
• Signature: $[28, 0]$
• Discriminant: $8.196\times 10^{47}$
• Root discriminant: $51.43$
• Ramified primes: $2, 29$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{28}$ (as 28T1)

Normalized defining polynomial

\( x^{28} - 29 x^{26} + 377 x^{24} - 2900 x^{22} + 14674 x^{20} - 51359 x^{18} + 127281 x^{16} - 224808 x^{14} + 281010 x^{12} - 243542 x^{10} + 140998 x^{8} - 51272 x^{6} + 10556 x^{4} - 1015 x^{2} + 29 \)

Invariants

Degree:		$28$
Signature:		$[28, 0]$
Discriminant:		\(819569564076950716311987772907236898045117333504\)\(\medspace = 2^{28}\cdot 29^{27}\)
Root discriminant:		$51.43$
Ramified primes:		$2, 29$
$|\Gal(K/\Q)|$:		$28$
This field is Galois and abelian over $\Q$.
Conductor:		\(116=2^{2}\cdot 29\)
Dirichlet character group:		$\lbrace$$\chi_{116}(1,·)$, $\chi_{116}(3,·)$, $\chi_{116}(5,·)$, $\chi_{116}(65,·)$, $\chi_{116}(9,·)$, $\chi_{116}(79,·)$, $\chi_{116}(11,·)$, $\chi_{116}(109,·)$, $\chi_{116}(13,·)$, $\chi_{116}(15,·)$, $\chi_{116}(81,·)$, $\chi_{116}(75,·)$, $\chi_{116}(19,·)$, $\chi_{116}(25,·)$, $\chi_{116}(27,·)$, $\chi_{116}(93,·)$, $\chi_{116}(31,·)$, $\chi_{116}(33,·)$, $\chi_{116}(99,·)$, $\chi_{116}(39,·)$, $\chi_{116}(43,·)$, $\chi_{116}(45,·)$, $\chi_{116}(47,·)$, $\chi_{116}(49,·)$, $\chi_{116}(53,·)$, $\chi_{116}(55,·)$, $\chi_{116}(57,·)$, $\chi_{116}(95,·)$$\rbrace$
This is not a CM field.

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

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

Class group and class number

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

Unit group

Rank:		$27$
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:		\( 822416597953191.9 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{28}\cdot(2\pi)^{0}\cdot 822416597953191.9 \cdot 1}{2\sqrt{819569564076950716311987772907236898045117333504}}\approx 0.121929512845963$ (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

\(\Q(\sqrt{29}) \), 4.4.390224.1, 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	R	$28$	${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$	$28$	${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$	$28$	${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$	R	$28$	$28$	${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$	$28$	$28$	${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
29	Data not computed