Global number field 18.0.119665574759900159778264689410048.1

• Label: 18.0.11966557475...0048.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,2^{18}\cdot 37^{17}$
• Root discriminant: $60.55$
• Ramified primes: $2, 37$
• Class number: $1526$ (GRH)
• Class group: $[1526]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} + 37 x^{16} + 518 x^{14} + 3515 x^{12} + 12284 x^{10} + 22200 x^{8} + 20683 x^{6} + 9287 x^{4} + 1591 x^{2} + 37 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-119665574759900159778264689410048=-\,2^{18}\cdot 37^{17}\)
Root discriminant:		$60.55$
Ramified primes:		$2, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(148=2^{2}\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{148}(1,·)$, $\chi_{148}(67,·)$, $\chi_{148}(9,·)$, $\chi_{148}(11,·)$, $\chi_{148}(115,·)$, $\chi_{148}(145,·)$, $\chi_{148}(3,·)$, $\chi_{148}(139,·)$, $\chi_{148}(27,·)$, $\chi_{148}(95,·)$, $\chi_{148}(33,·)$, $\chi_{148}(99,·)$, $\chi_{148}(81,·)$, $\chi_{148}(49,·)$, $\chi_{148}(147,·)$, $\chi_{148}(53,·)$, $\chi_{148}(137,·)$, $\chi_{148}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{43} a^{14} + \frac{14}{43} a^{10} - \frac{13}{43} a^{8} - \frac{10}{43} a^{6} + \frac{11}{43} a^{4} - \frac{11}{43} a^{2} - \frac{21}{43}$, $\frac{1}{43} a^{15} + \frac{14}{43} a^{11} - \frac{13}{43} a^{9} - \frac{10}{43} a^{7} + \frac{11}{43} a^{5} - \frac{11}{43} a^{3} - \frac{21}{43} a$, $\frac{1}{2769316831} a^{16} + \frac{31484785}{2769316831} a^{14} - \frac{1186318945}{2769316831} a^{12} + \frac{28402830}{2769316831} a^{10} + \frac{454901551}{2769316831} a^{8} + \frac{552171309}{2769316831} a^{6} - \frac{1045941801}{2769316831} a^{4} - \frac{852680188}{2769316831} a^{2} - \frac{7266499}{2769316831}$, $\frac{1}{2769316831} a^{17} + \frac{31484785}{2769316831} a^{15} - \frac{1186318945}{2769316831} a^{13} + \frac{28402830}{2769316831} a^{11} + \frac{454901551}{2769316831} a^{9} + \frac{552171309}{2769316831} a^{7} - \frac{1045941801}{2769316831} a^{5} - \frac{852680188}{2769316831} a^{3} - \frac{7266499}{2769316831} a$

Class group and class number

$C_{1526}$, which has order $1526$ (assuming GRH)

Unit group

Rank:		$8$
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:		\( 409151.310213 \) (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

\(\Q(\sqrt{-37}) \), 3.3.1369.1, 6.0.4438013248.1, 9.9.3512479453921.1

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	R	$18$	$18$	$18$	${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$	$18$	$18$	${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$	R	${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
37	Data not computed