Number field 27.27.706965049015104706497203195837614914543357369.1: \(\Q(\zeta_{81})^+\)

• Label: 27.27.706...369.1
• Degree: $27$
• Signature: $[27, 0]$
• Discriminant: $7.070\times 10^{44}$
• Root discriminant: \(45.82\)
• Ramified prime: $3$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{27}$ (as 27T1)

Normalized defining polynomial

x^27 - 27*x^25 + 324*x^23 - 2277*x^21 + 10395*x^19 - 32319*x^17 + 69768*x^15 - 104652*x^13 + 107406*x^11 - 72930*x^9 + 30888*x^7 - 7371*x^5 + 819*x^3 - 27*x - 1

Invariants

Degree:		$27$
Signature:		$[27, 0]$
Discriminant:		\(706965049015104706497203195837614914543357369\) \(\medspace = 3^{94}\)
Root discriminant:		\(45.82\)
Galois root discriminant:		$3^{94/27}\approx 45.82355857897648$
Ramified primes:		\(3\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$27$
This field is Galois and abelian over $\Q$.
Conductor:		\(81=3^{4}\)
Dirichlet character group:		$\lbrace$$\chi_{81}(64,·)$, $\chi_{81}(1,·)$, $\chi_{81}(67,·)$, $\chi_{81}(4,·)$, $\chi_{81}(70,·)$, $\chi_{81}(7,·)$, $\chi_{81}(73,·)$, $\chi_{81}(10,·)$, $\chi_{81}(76,·)$, $\chi_{81}(13,·)$, $\chi_{81}(79,·)$, $\chi_{81}(16,·)$, $\chi_{81}(19,·)$, $\chi_{81}(22,·)$, $\chi_{81}(25,·)$, $\chi_{81}(28,·)$, $\chi_{81}(31,·)$, $\chi_{81}(34,·)$, $\chi_{81}(37,·)$, $\chi_{81}(40,·)$, $\chi_{81}(43,·)$, $\chi_{81}(46,·)$, $\chi_{81}(49,·)$, $\chi_{81}(52,·)$, $\chi_{81}(55,·)$, $\chi_{81}(58,·)$, $\chi_{81}(61,·)$$\rbrace$
This is not a CM field.

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

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

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

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

Unit group

Rank:		$26$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}-a^{6}+2079a^{5}+6a^{4}-385a^{3}-9a^{2}+21a+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{3}-3a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}-a^{8}+2508a^{7}+8a^{6}-1254a^{5}-20a^{4}+285a^{3}+16a^{2}-19a-2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{5}-5a^{3}+5a$, $a^{17}-17a^{15}+119a^{13}-442a^{11}-a^{10}+935a^{9}+10a^{8}-1122a^{7}-35a^{6}+714a^{5}+50a^{4}-204a^{3}-25a^{2}+17a+2$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}-a^{4}+506a^{3}+4a^{2}-23a-2$, $a^{4}-4a^{2}+2$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}-a^{2}+25a+2$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-a^{7}-2640a^{6}+7a^{5}+825a^{4}-14a^{3}-100a^{2}+7a+2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{10}-10a^{8}+35a^{6}-a^{5}-50a^{4}+5a^{3}+25a^{2}-5a-1$ (assuming GRH)
Regulator:		\( 68981171346332.37 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{27}\cdot(2\pi)^{0}\cdot 68981171346332.37 \cdot 1}{2\cdot\sqrt{706965049015104706497203195837614914543357369}}\cr\approx \mathstrut & 0.174105094869797 \end{aligned}\] (assuming GRH)

Galois group

$C_{27}$ (as 27T1):

A cyclic group of order 27

The 27 conjugacy class representatives for $C_{27}$

Character table for $C_{27}$

Intermediate fields

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\)

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	$27$	R	$27$	$27$	$27$	$27$	${\href{/padicField/17.9.0.1}{9} }^{3}$	${\href{/padicField/19.9.0.1}{9} }^{3}$	$27$	$27$	$27$	${\href{/padicField/37.9.0.1}{9} }^{3}$	$27$	$27$	$27$	${\href{/padicField/53.3.0.1}{3} }^{9}$	$27$

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
\(3\)		Deg $27$	$27$	$1$	$94$