Global number field 18.0.232218265089212416000000000.1

• Label: 18.0.23221826508...0000.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,2^{33}\cdot 5^{9}\cdot 7^{12}$
• Root discriminant: $29.16$
• Ramified primes: $2, 5, 7$
• Class number: $18$ (GRH)
• Class group: $[3, 6]$ (GRH)
• Galois group: $S_3 \times C_6$ (as 18T6)

Normalized defining polynomial

\( x^{18} + 14 x^{16} + 77 x^{14} + 310 x^{12} + 994 x^{10} + 1876 x^{8} + 3289 x^{6} + 3290 x^{4} + 1400 x^{2} + 1000 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-232218265089212416000000000=-\,2^{33}\cdot 5^{9}\cdot 7^{12}\)
Root discriminant:		$29.16$
Ramified primes:		$2, 5, 7$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{3}{11} a^{10} + \frac{3}{11} a^{8} - \frac{1}{11} a^{6} - \frac{3}{11} a^{2} + \frac{4}{11}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{11} + \frac{3}{11} a^{9} - \frac{1}{11} a^{7} - \frac{3}{11} a^{3} + \frac{4}{11} a$, $\frac{1}{990} a^{14} - \frac{1}{165} a^{12} - \frac{263}{990} a^{10} + \frac{32}{99} a^{8} + \frac{7}{495} a^{6} - \frac{172}{495} a^{4} - \frac{47}{330} a^{2} + \frac{1}{99}$, $\frac{1}{990} a^{15} - \frac{1}{165} a^{13} - \frac{263}{990} a^{11} + \frac{32}{99} a^{9} + \frac{7}{495} a^{7} - \frac{172}{495} a^{5} - \frac{47}{330} a^{3} + \frac{1}{99} a$, $\frac{1}{163233605700} a^{16} + \frac{11622151}{40808401425} a^{14} + \frac{6390397837}{163233605700} a^{12} + \frac{34586329}{247323645} a^{10} - \frac{17925612403}{81616802850} a^{8} + \frac{657454678}{13602800475} a^{6} - \frac{15733188071}{163233605700} a^{4} + \frac{570863272}{1632336057} a^{2} + \frac{507209797}{1632336057}$, $\frac{1}{163233605700} a^{17} + \frac{11622151}{40808401425} a^{15} + \frac{6390397837}{163233605700} a^{13} + \frac{34586329}{247323645} a^{11} - \frac{17925612403}{81616802850} a^{9} + \frac{657454678}{13602800475} a^{7} - \frac{15733188071}{163233605700} a^{5} + \frac{570863272}{1632336057} a^{3} + \frac{507209797}{1632336057} a$

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:		\( 34883.98422806271 \) (assuming GRH)

Galois group

$C_6\times S_3$ (as 18T6):

A solvable group of order 36

The 18 conjugacy class representatives for $S_3 \times C_6$

Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-10}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.0.614656000.1, 6.0.153664000.1, 9.3.941192000.1

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

Sibling fields

Galois closure:	data not computed
Degree 12 sibling:	data not computed
Degree 18 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	${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$	R	R	${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.9.7	$x^{6} + 4 x^{4} + 4 x^{2} - 24$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.12.24.318	$x^{12} + 60 x^{11} + 14 x^{10} + 36 x^{9} - 34 x^{8} - 32 x^{7} - 48 x^{6} - 32 x^{5} + 36 x^{4} - 16 x^{3} - 40 x^{2} - 48 x + 56$	$4$	$3$	$24$	$C_6\times C_2$	$[2, 3]^{3}$
$5$	5.6.3.2	$x^{6} - 25 x^{2} + 250$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.12.6.1	$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$7$	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	$x^{9} + 42 x^{6} + 539 x^{3} + 2744$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e3_5.2t1.2c1	$1$	$ 2^{3} \cdot 5 $	$x^{2} + 10$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_5.2t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{2} + 5$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
	1.2e3_7.6t1.2c1	$1$	$ 2^{3} \cdot 7 $	$x^{6} - 10 x^{4} + 24 x^{2} - 8$	$C_6$ (as 6T1)	$0$	$1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
	1.2e2_5_7.6t1.2c1	$1$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281$	$C_6$ (as 6T1)	$0$	$-1$
*	1.2e3_5_7.6t1.3c1	$1$	$ 2^{3} \cdot 5 \cdot 7 $	$x^{6} - 2 x^{5} + 27 x^{4} - 34 x^{3} + 322 x^{2} - 224 x + 1561$	$C_6$ (as 6T1)	$0$	$-1$
	1.2e2_5_7.6t1.2c2	$1$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281$	$C_6$ (as 6T1)	$0$	$-1$
	1.2e3_7.6t1.2c2	$1$	$ 2^{3} \cdot 7 $	$x^{6} - 10 x^{4} + 24 x^{2} - 8$	$C_6$ (as 6T1)	$0$	$1$
*	1.2e3_5_7.6t1.3c2	$1$	$ 2^{3} \cdot 5 \cdot 7 $	$x^{6} - 2 x^{5} + 27 x^{4} - 34 x^{3} + 322 x^{2} - 224 x + 1561$	$C_6$ (as 6T1)	$0$	$-1$
*	2.2e2_5_7e2.3t2.1c1	$2$	$ 2^{2} \cdot 5 \cdot 7^{2}$	$x^{3} - x^{2} + 5 x - 13$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e6_5_7e2.6t3.5c1	$2$	$ 2^{6} \cdot 5 \cdot 7^{2}$	$x^{6} - 6 x^{4} - 16 x^{2} + 160$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.2e6_5_7.12t18.2c1	$2$	$ 2^{6} \cdot 5 \cdot 7 $	$x^{18} + 14 x^{16} + 77 x^{14} + 310 x^{12} + 994 x^{10} + 1876 x^{8} + 3289 x^{6} + 3290 x^{4} + 1400 x^{2} + 1000$	$S_3 \times C_6$ (as 18T6)	$0$	$0$
*	2.2e2_5_7.6t5.1c1	$2$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{6} - 2 x^{5} + 3 x^{4} + 4 x^{2} + 2 x + 1$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.2e6_5_7.12t18.2c2	$2$	$ 2^{6} \cdot 5 \cdot 7 $	$x^{18} + 14 x^{16} + 77 x^{14} + 310 x^{12} + 994 x^{10} + 1876 x^{8} + 3289 x^{6} + 3290 x^{4} + 1400 x^{2} + 1000$	$S_3 \times C_6$ (as 18T6)	$0$	$0$
*	2.2e2_5_7.6t5.1c2	$2$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{6} - 2 x^{5} + 3 x^{4} + 4 x^{2} + 2 x + 1$	$S_3\times C_3$ (as 6T5)	$0$	$0$

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 *.