Number field 18.0.2954312706550833698643.1: \(\Q(\zeta_{27})\)

• Label: 18.0.295...643.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-2.954\times 10^{21}$
• Root discriminant: $15.59$
• Ramified prime: $3$
• Class number: $1$
• Class group: trivial
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\(x^{18} - x^{9} + 1\)  

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-2954312706550833698643\)\(\medspace = -\,3^{45}\)
Root discriminant:		$15.59$
Ramified primes:		$3$
$\card{ \Gal(K/\Q) }$:		$18$
This field is Galois and abelian over $\Q$.
Conductor:		\(27=3^{3}\)
Dirichlet character group:		$\lbrace$$\chi_{27}(1,·)$, $\chi_{27}(2,·)$, $\chi_{27}(4,·)$, $\chi_{27}(5,·)$, $\chi_{27}(7,·)$, $\chi_{27}(8,·)$, $\chi_{27}(10,·)$, $\chi_{27}(11,·)$, $\chi_{27}(13,·)$, $\chi_{27}(14,·)$, $\chi_{27}(16,·)$, $\chi_{27}(17,·)$, $\chi_{27}(19,·)$, $\chi_{27}(20,·)$, $\chi_{27}(22,·)$, $\chi_{27}(23,·)$, $\chi_{27}(25,·)$, $\chi_{27}(26,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$  

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$8$
Torsion generator:		\( a \) (order $54$)
Fundamental units:		\( a^{13} + a \),  \( a^{15} - a^{9} + 1 \),  \( a^{14} + a^{9} - a^{5} \),  \( a^{13} + a^{9} \),  \( a^{8} - a^{7} \),  \( a^{16} + a^{9} - a^{7} \),  \( a^{4} + a^{2} \),  \( a^{10} + a^{9} - a \)
Regulator:		\( 40934.0329443 \)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{9}\cdot 40934.0329443 \cdot 1}{54\sqrt{2954312706550833698643}}\approx 0.212853776024$

Galois group

$C_{18}$ (as 18T1):

A cyclic group of order 18

The 18 conjugacy class representatives for $C_{18}$

Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), \(\Q(\zeta_{27})^+\)

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	$18$	R	$18$	${\href{/padicField/7.9.0.1}{9} }^{2}$	$18$	${\href{/padicField/13.9.0.1}{9} }^{2}$	${\href{/padicField/17.6.0.1}{6} }^{3}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	$18$	$18$	${\href{/padicField/31.9.0.1}{9} }^{2}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	$18$	${\href{/padicField/43.9.0.1}{9} }^{2}$	$18$	${\href{/padicField/53.2.0.1}{2} }^{9}$	$18$

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 $18$	$18$	$1$	$45$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.3.2t1.a.a	$1$	$ 3 $	\(\Q(\sqrt{-3}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.27.9t1.a.a	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*	1.27.18t1.a.a	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.27.9t1.a.b	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*	1.27.18t1.a.b	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.9.3t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
*	1.9.6t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})\)	$C_6$ (as 6T1)	$0$	$-1$
*	1.27.9t1.a.c	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*	1.27.18t1.a.c	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.27.9t1.a.d	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*	1.27.18t1.a.d	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.9.3t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
*	1.9.6t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})\)	$C_6$ (as 6T1)	$0$	$-1$
*	1.27.9t1.a.e	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*	1.27.18t1.a.e	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.27.9t1.a.f	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*	1.27.18t1.a.f	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.