Global number field 12.0.2758547353515625.1

• Label: 12.0.2758547353515625.1
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $5^{10}\cdot 7^{10}$
• Root discriminant: $19.35$
• Ramified primes: $5, 7$
• Class number: $3$
• Class group: $[3]$
• Galois group: $C_6\times S_3$ (as 12T18)

Normalized defining polynomial

\( x^{12} - 5 x^{11} + 13 x^{10} - 40 x^{9} + 100 x^{8} - 195 x^{7} + 370 x^{6} - 590 x^{5} + 855 x^{4} - 870 x^{3} + 908 x^{2} - 400 x + 64 \)

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(2758547353515625=5^{10}\cdot 7^{10}\)
Root discriminant:		$19.35$
Ramified primes:		$5, 7$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{44} a^{10} + \frac{9}{44} a^{9} - \frac{9}{44} a^{8} - \frac{1}{22} a^{7} - \frac{1}{11} a^{6} + \frac{1}{44} a^{5} + \frac{2}{11} a^{4} - \frac{5}{22} a^{3} + \frac{15}{44} a^{2} - \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{166720117168} a^{11} - \frac{963290513}{166720117168} a^{10} + \frac{30207277985}{166720117168} a^{9} + \frac{10512671693}{41680029292} a^{8} + \frac{13761489167}{41680029292} a^{7} + \frac{9078468109}{166720117168} a^{6} + \frac{28457837843}{83360058584} a^{5} - \frac{2857206173}{7578187144} a^{4} + \frac{50006106191}{166720117168} a^{3} + \frac{271572537}{7578187144} a^{2} - \frac{11414512861}{41680029292} a - \frac{3588834424}{10420007323}$

Class group and class number

$C_{3}$, which has order $3$

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
Regulator:		\( 357.383618201 \)

Galois group

$C_6\times S_3$ (as 12T18):

A solvable group of order 36

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

Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.10504375.1

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 18 siblings:	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	${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$	R	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$5$	5.12.10.2	$x^{12} + 15 x^{6} + 100$	$6$	$2$	$10$	$C_6\times S_3$	$[\ ]_{6}^{6}$
$7$	7.12.10.1	$x^{12} - 70 x^{6} + 35721$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5_7.2t1.1c1	$1$	$ 5 \cdot 7 $	$x^{2} - x + 9$	$C_2$ (as 2T1)	$1$	$-1$
	1.7.6t1.1c1	$1$	$ 7 $	$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$	$C_6$ (as 6T1)	$0$	$-1$
	1.5_7.6t1.2c1	$1$	$ 5 \cdot 7 $	$x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$	$C_6$ (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)	$0$	$1$
	1.5_7.6t1.1c1	$1$	$ 5 \cdot 7 $	$x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$	$C_6$ (as 6T1)	$0$	$1$
	1.7.6t1.1c2	$1$	$ 7 $	$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$	$C_6$ (as 6T1)	$0$	$-1$
	1.5_7.6t1.1c2	$1$	$ 5 \cdot 7 $	$x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$	$C_6$ (as 6T1)	$0$	$1$
	1.5_7.6t1.2c2	$1$	$ 5 \cdot 7 $	$x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$	$C_6$ (as 6T1)	$0$	$-1$
	2.5e2_7.3t2.1c1	$2$	$ 5^{2} \cdot 7 $	$x^{3} - x^{2} + 2 x - 3$	$S_3$ (as 3T2)	$1$	$0$
	2.5e2_7.6t3.1c1	$2$	$ 5^{2} \cdot 7 $	$x^{6} - x^{5} + 4 x + 1$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.5e2_7e2.6t5.1c1	$2$	$ 5^{2} \cdot 7^{2}$	$x^{6} - x^{5} + x^{4} + 6 x^{3} + 8 x^{2} - 29 x + 22$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.5e2_7e2.6t5.1c2	$2$	$ 5^{2} \cdot 7^{2}$	$x^{6} - x^{5} + x^{4} + 6 x^{3} + 8 x^{2} - 29 x + 22$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.5e2_7e2.12t18.2c1	$2$	$ 5^{2} \cdot 7^{2}$	$x^{12} - 5 x^{11} + 13 x^{10} - 40 x^{9} + 100 x^{8} - 195 x^{7} + 370 x^{6} - 590 x^{5} + 855 x^{4} - 870 x^{3} + 908 x^{2} - 400 x + 64$	$C_6\times S_3$ (as 12T18)	$0$	$0$
*	2.5e2_7e2.12t18.2c2	$2$	$ 5^{2} \cdot 7^{2}$	$x^{12} - 5 x^{11} + 13 x^{10} - 40 x^{9} + 100 x^{8} - 195 x^{7} + 370 x^{6} - 590 x^{5} + 855 x^{4} - 870 x^{3} + 908 x^{2} - 400 x + 64$	$C_6\times S_3$ (as 12T18)	$0$	$0$

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 *.