Global number field 18.0.38477475727151575852122112.1

• Label: 18.0.38477475727...2112.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,2^{18}\cdot 7^{12}\cdot 13^{9}$
• Root discriminant: $26.39$
• Ramified primes: $2, 7, 13$
• Class number: $8$
• Class group: $[2, 2, 2]$
• Galois group: $S_3 \times C_3$ (as 18T3)

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 16 x^{16} - 22 x^{15} + 43 x^{14} - 16 x^{13} + 254 x^{12} - 1038 x^{11} + 4438 x^{10} - 14158 x^{9} + 33530 x^{8} - 63172 x^{7} + 96066 x^{6} - 118196 x^{5} + 113867 x^{4} - 81796 x^{3} + 41236 x^{2} - 13182 x + 2197 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-38477475727151575852122112=-\,2^{18}\cdot 7^{12}\cdot 13^{9}\)
Root discriminant:		$26.39$
Ramified primes:		$2, 7, 13$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{3}{13} a^{11} + \frac{2}{13} a^{10} + \frac{4}{13} a^{9} + \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{5}{13} a^{6} - \frac{5}{13} a^{5} + \frac{2}{13} a^{4} + \frac{6}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{12} + \frac{3}{26} a^{11} + \frac{9}{26} a^{10} + \frac{11}{26} a^{9} + \frac{3}{26} a^{8} - \frac{7}{26} a^{7} - \frac{11}{26} a^{6} + \frac{9}{26} a^{5} + \frac{11}{26} a^{4} + \frac{3}{26} a^{3} + \frac{9}{26} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{52} a^{14} - \frac{1}{26} a^{12} - \frac{1}{2} a^{11} - \frac{7}{26} a^{10} + \frac{6}{13} a^{9} - \frac{2}{13} a^{8} - \frac{5}{13} a^{7} - \frac{2}{13} a^{6} + \frac{7}{26} a^{5} + \frac{3}{26} a^{4} - \frac{3}{13} a^{3} + \frac{9}{26} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{104} a^{15} - \frac{1}{104} a^{14} - \frac{1}{52} a^{13} - \frac{5}{26} a^{11} + \frac{17}{52} a^{10} - \frac{5}{13} a^{9} - \frac{5}{13} a^{8} + \frac{3}{13} a^{7} - \frac{23}{52} a^{6} - \frac{3}{13} a^{5} + \frac{15}{52} a^{4} - \frac{17}{52} a^{3} - \frac{5}{26} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{1024816} a^{16} + \frac{201}{128102} a^{15} - \frac{1219}{1024816} a^{14} - \frac{7681}{512408} a^{13} + \frac{4551}{256204} a^{12} + \frac{36483}{512408} a^{11} - \frac{67447}{512408} a^{10} - \frac{29487}{128102} a^{9} + \frac{10679}{64051} a^{8} - \frac{174155}{512408} a^{7} - \frac{99351}{512408} a^{6} + \frac{65667}{512408} a^{5} - \frac{95915}{256204} a^{4} + \frac{4393}{39416} a^{3} - \frac{28845}{78832} a^{2} - \frac{1119}{3032} a + \frac{815}{6064}$, $\frac{1}{9747391831660495461798752} a^{17} + \frac{1221794478731210899}{9747391831660495461798752} a^{16} - \frac{3631681346468108093387}{9747391831660495461798752} a^{15} - \frac{32518814475531479542355}{9747391831660495461798752} a^{14} - \frac{199464490972374864425}{374899685833095979299952} a^{13} - \frac{152429446509733754616811}{4873695915830247730899376} a^{12} + \frac{720530677054421514183457}{2436847957915123865449688} a^{11} - \frac{2039083864271852504247657}{4873695915830247730899376} a^{10} - \frac{560928968903383880230793}{1218423978957561932724844} a^{9} - \frac{1599882404699414371021411}{4873695915830247730899376} a^{8} - \frac{114965188992394276239811}{609211989478780966362422} a^{7} + \frac{1190158601523608909815387}{2436847957915123865449688} a^{6} - \frac{712350588038654686487125}{4873695915830247730899376} a^{5} + \frac{2259296842176507160182131}{4873695915830247730899376} a^{4} - \frac{316099244911798989307639}{749799371666191958599904} a^{3} - \frac{242268813716812366925029}{749799371666191958599904} a^{2} + \frac{4445187778324608243525}{57676874743553227584608} a + \frac{20423728472749678183013}{57676874743553227584608}$

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

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
Regulator:		\( 39919.4689246 \)

Galois group

$C_3\times S_3$ (as 18T3):

A solvable group of order 18

The 9 conjugacy class representatives for $S_3 \times C_3$

Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-13}) \), 3.1.2548.1 x3, \(\Q(\zeta_{7})^+\), 6.0.337599808.2, 6.0.6889792.1 x2, 6.0.337599808.1, 9.3.16542390592.2 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling:	6.0.6889792.1
Degree 9 sibling:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$	R	${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$	2.6.6.5	$x^{6} - 2 x^{4} + x^{2} - 3$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.5	$x^{6} - 2 x^{4} + x^{2} - 3$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.5	$x^{6} - 2 x^{4} + x^{2} - 3$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$7$	7.9.6.1	$x^{9} + 42 x^{6} + 539 x^{3} + 2744$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
	7.9.6.1	$x^{9} + 42 x^{6} + 539 x^{3} + 2744$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$13$	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_13.2t1.1c1	$1$	$ 2^{2} \cdot 13 $	$x^{2} + 13$	$C_2$ (as 2T1)	$1$	$-1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_7_13.6t1.8c1	$1$	$ 2^{2} \cdot 7 \cdot 13 $	$x^{6} - 2 x^{5} + 36 x^{4} - 46 x^{3} + 535 x^{2} - 368 x + 3121$	$C_6$ (as 6T1)	$0$	$-1$
*	1.2e2_7_13.6t1.8c2	$1$	$ 2^{2} \cdot 7 \cdot 13 $	$x^{6} - 2 x^{5} + 36 x^{4} - 46 x^{3} + 535 x^{2} - 368 x + 3121$	$C_6$ (as 6T1)	$0$	$-1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*^2	2.2e2_7e2_13.3t2.1c1	$2$	$ 2^{2} \cdot 7^{2} \cdot 13 $	$x^{3} - x^{2} + 12 x + 8$	$S_3$ (as 3T2)	$1$	$0$
*^2	2.2e2_7_13.6t5.3c1	$2$	$ 2^{2} \cdot 7 \cdot 13 $	$x^{18} - 4 x^{17} + 16 x^{16} - 22 x^{15} + 43 x^{14} - 16 x^{13} + 254 x^{12} - 1038 x^{11} + 4438 x^{10} - 14158 x^{9} + 33530 x^{8} - 63172 x^{7} + 96066 x^{6} - 118196 x^{5} + 113867 x^{4} - 81796 x^{3} + 41236 x^{2} - 13182 x + 2197$	$S_3 \times C_3$ (as 18T3)	$0$	$0$
*^2	2.2e2_7_13.6t5.3c2	$2$	$ 2^{2} \cdot 7 \cdot 13 $	$x^{18} - 4 x^{17} + 16 x^{16} - 22 x^{15} + 43 x^{14} - 16 x^{13} + 254 x^{12} - 1038 x^{11} + 4438 x^{10} - 14158 x^{9} + 33530 x^{8} - 63172 x^{7} + 96066 x^{6} - 118196 x^{5} + 113867 x^{4} - 81796 x^{3} + 41236 x^{2} - 13182 x + 2197$	$S_3 \times C_3$ (as 18T3)	$0$	$0$

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 *.