Global number field 9.3.2063322368402112.1

• Label: 9.3.2063322368402112.1
• Degree: $9$
• Signature: $[3, 3]$
• Discriminant: $-\,2^{6}\cdot 3^{3}\cdot 103^{6}$
• Root discriminant: $50.31$
• Ramified primes: $2, 3, 103$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $S_3\times C_3$ (as 9T4)

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 4 x^{7} + 14 x^{6} - 170 x^{5} + 440 x^{4} + 24 x^{3} - 1834 x^{2} + 2733 x - 993 \)

Invariants

Degree:		$9$
Signature:		$[3, 3]$
Discriminant:		\(-2063322368402112=-\,2^{6}\cdot 3^{3}\cdot 103^{6}\)
Root discriminant:		$50.31$
Ramified primes:		$2, 3, 103$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{6} + \frac{1}{12} a^{4} - \frac{5}{12} a^{2} + \frac{1}{4}$, $\frac{1}{72} a^{7} - \frac{1}{72} a^{6} + \frac{1}{72} a^{5} + \frac{11}{72} a^{4} + \frac{7}{72} a^{3} - \frac{7}{72} a^{2} + \frac{3}{8} a + \frac{11}{24}$, $\frac{1}{98856} a^{8} - \frac{11}{32952} a^{7} + \frac{263}{10984} a^{6} + \frac{197}{10984} a^{5} + \frac{7385}{32952} a^{4} - \frac{727}{10984} a^{3} - \frac{41215}{98856} a^{2} - \frac{11803}{32952} a - \frac{419}{4119}$

Class group and class number

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

Unit group

Rank:		$5$
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:		\( 99818.3236792 \) (assuming GRH)

Galois group

$C_3\times S_3$ (as 9T4):

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

3.3.10609.1, 3.1.127308.1

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 6 sibling:	6.0.4583088.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$	${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$	${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$	${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$	${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$	2.9.6.1	$x^{9} - 4 x^{3} + 8$	$3$	$3$	$6$	$S_3\times C_3$	$[\ ]_{3}^{6}$
$3$	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$103$	103.9.6.1	$x^{9} + 8961 x^{6} + 26755898 x^{3} + 26650518803$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.103.3t1.1c1	$1$	$ 103 $	$x^{3} - x^{2} - 34 x + 61$	$C_3$ (as 3T1)	$0$	$1$
	1.3_103.6t1.1c1	$1$	$ 3 \cdot 103 $	$x^{6} - x^{5} + 35 x^{4} - 88 x^{3} + 1217 x^{2} - 2074 x + 3721$	$C_6$ (as 6T1)	$0$	$-1$
	1.3_103.6t1.1c2	$1$	$ 3 \cdot 103 $	$x^{6} - x^{5} + 35 x^{4} - 88 x^{3} + 1217 x^{2} - 2074 x + 3721$	$C_6$ (as 6T1)	$0$	$-1$
*	1.103.3t1.1c2	$1$	$ 103 $	$x^{3} - x^{2} - 34 x + 61$	$C_3$ (as 3T1)	$0$	$1$
*	2.2e2_3_103e2.3t2.1c1	$2$	$ 2^{2} \cdot 3 \cdot 103^{2}$	$x^{3} - 206$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_3_103.6t5.2c1	$2$	$ 2^{2} \cdot 3 \cdot 103 $	$x^{9} - 3 x^{8} + 4 x^{7} + 14 x^{6} - 170 x^{5} + 440 x^{4} + 24 x^{3} - 1834 x^{2} + 2733 x - 993$	$S_3\times C_3$ (as 9T4)	$0$	$0$
*	2.2e2_3_103.6t5.2c2	$2$	$ 2^{2} \cdot 3 \cdot 103 $	$x^{9} - 3 x^{8} + 4 x^{7} + 14 x^{6} - 170 x^{5} + 440 x^{4} + 24 x^{3} - 1834 x^{2} + 2733 x - 993$	$S_3\times C_3$ (as 9T4)	$0$	$0$

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 *.