Global number field 18.18.735565072612935262326166126592.1

• Label: 18.18.7355650726...6592.1
• Degree: $18$
• Signature: $[18, 0]$
• Discriminant: $2^{27}\cdot 19^{17}$
• Root discriminant: $45.63$
• Ramified primes: $2, 19$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 38 x^{16} + 608 x^{14} - 5320 x^{12} + 27664 x^{10} - 86944 x^{8} + 160512 x^{6} - 160512 x^{4} + 72960 x^{2} - 9728 \)

Invariants

Degree:		$18$
Signature:		$[18, 0]$
Discriminant:		\(735565072612935262326166126592=2^{27}\cdot 19^{17}\)
Root discriminant:		$45.63$
Ramified primes:		$2, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(152=2^{3}\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{152}(1,·)$, $\chi_{152}(67,·)$, $\chi_{152}(147,·)$, $\chi_{152}(9,·)$, $\chi_{152}(75,·)$, $\chi_{152}(17,·)$, $\chi_{152}(3,·)$, $\chi_{152}(137,·)$, $\chi_{152}(25,·)$, $\chi_{152}(91,·)$, $\chi_{152}(27,·)$, $\chi_{152}(81,·)$, $\chi_{152}(107,·)$, $\chi_{152}(49,·)$, $\chi_{152}(51,·)$, $\chi_{152}(73,·)$, $\chi_{152}(121,·)$, $\chi_{152}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$

Class group and class number

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

Unit group

Rank:		$17$
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:		\( 1274691143.44 \) (assuming GRH)

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

\(\Q(\sqrt{38}) \), 3.3.361.1, 6.6.1267762688.1, \(\Q(\zeta_{19})^+\)

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	$18$	$18$	${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$	R	$18$	${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$	$18$	${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$	$18$

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
2	Data not computed
19	Data not computed