Number field 30.30.17581401814197409148890873176573567284303576419397.1: \(\Q(\zeta_{77})^+\)

• Label: 30.30.175...397.1
• Degree: $30$
• Signature: $[30, 0]$
• Discriminant: $1.758\times 10^{49}$
• Root discriminant: $43.80$
• Ramified primes: $7, 11$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

\( x^{30} - x^{29} - 30 x^{28} + 29 x^{27} + 405 x^{26} - 377 x^{25} - 3250 x^{24} + 2901 x^{23} + 17249 x^{22} - 14697 x^{21} - 63734 x^{20} + 51590 x^{19} + 168035 x^{18} - 128611 x^{17} - 318629 x^{16} + 229651 x^{15} + 432159 x^{14} - 292608 x^{13} - 411241 x^{12} + 261924 x^{11} + 264472 x^{10} - 159873 x^{9} - 107406 x^{8} + 63143 x^{7} + 24051 x^{6} - 14609 x^{5} - 2010 x^{4} + 1562 x^{3} - 72 x^{2} - 24 x + 1 \)

Invariants

Degree:		$30$
Signature:		$[30, 0]$
Discriminant:		\(17581401814197409148890873176573567284303576419397\)\(\medspace = 7^{25}\cdot 11^{27}\)
Root discriminant:		$43.80$
Ramified primes:		$7, 11$
$|\Gal(K/\Q)|$:		$30$
This field is Galois and abelian over $\Q$.
Conductor:		\(77=7\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(67,·)$, $\chi_{77}(4,·)$, $\chi_{77}(6,·)$, $\chi_{77}(71,·)$, $\chi_{77}(9,·)$, $\chi_{77}(10,·)$, $\chi_{77}(76,·)$, $\chi_{77}(13,·)$, $\chi_{77}(15,·)$, $\chi_{77}(16,·)$, $\chi_{77}(17,·)$, $\chi_{77}(19,·)$, $\chi_{77}(23,·)$, $\chi_{77}(24,·)$, $\chi_{77}(25,·)$, $\chi_{77}(68,·)$, $\chi_{77}(36,·)$, $\chi_{77}(37,·)$, $\chi_{77}(40,·)$, $\chi_{77}(41,·)$, $\chi_{77}(52,·)$, $\chi_{77}(53,·)$, $\chi_{77}(54,·)$, $\chi_{77}(73,·)$, $\chi_{77}(58,·)$, $\chi_{77}(60,·)$, $\chi_{77}(61,·)$, $\chi_{77}(62,·)$$\rbrace$
This is not a CM field.

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

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

Class group and class number

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

Unit group

Rank:		$29$
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:		\( 864355592506536.0 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 864355592506536.0 \cdot 1}{2\sqrt{17581401814197409148890873176573567284303576419397}}\approx 0.110671442530690$ (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

\(\Q(\sqrt{77}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.22370117.1, 10.10.39630026842637.1, 15.15.886528337182930278529.1

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	$30$	$30$	$30$	R	R	${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$	$15^{2}$	$15^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$	$30$	$15^{2}$	${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$	$30$	$15^{2}$	$30$

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
7	Data not computed
11	Data not computed