Global number field 6.0.337599808.1

• Label: 6.0.337599808.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 7^{4}\cdot 13^{3}$
• Root discriminant: $26.39$
• Ramified primes: $2, 7, 13$
• Class number: $24$
• Class group: $[2, 2, 6]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 36 x^{4} - 46 x^{3} + 535 x^{2} - 368 x + 3121 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-337599808=-\,2^{6}\cdot 7^{4}\cdot 13^{3}\)
Root discriminant:		$26.39$
Ramified primes:		$2, 7, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(51,·)$, $\chi_{364}(261,·)$, $\chi_{364}(155,·)$, $\chi_{364}(53,·)$, $\chi_{364}(207,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{181061} a^{5} + \frac{78310}{181061} a^{4} + \frac{76686}{181061} a^{3} + \frac{2738}{181061} a^{2} + \frac{42567}{181061} a - \frac{7535}{181061}$

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{222}{181061} a^{5} + \frac{2964}{181061} a^{4} + \frac{4558}{181061} a^{3} + \frac{64653}{181061} a^{2} + \frac{34702}{181061} a + \frac{681023}{181061} \),  \( \frac{104}{181061} a^{5} - \frac{3505}{181061} a^{4} + \frac{8660}{181061} a^{3} - \frac{77370}{181061} a^{2} + \frac{81504}{181061} a - \frac{602579}{181061} \)
Regulator:		\( 2.10181872849 \)

Galois group

$C_6$ (as 6T1):

A cyclic group of order 6

The 6 conjugacy class representatives for $C_6$

Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-13}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{-13}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.6.0.1}{6} }$	${\href{/LocalNumberField/5.6.0.1}{6} }$	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.6.5	$x^{6} - 2 x^{4} + x^{2} - 3$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$7$	7.3.2.2	$x^{3} - 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.2	$x^{3} - 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$13$	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	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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)	$0$	$-1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	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)	$0$	$-1$

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 *.