Global number field 6.0.594663237952.6

• Label: 6.0.594663237952.6
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 7^{4}\cdot 157^{3}$
• Root discriminant: $91.70$
• Ramified primes: $2, 7, 157$
• Class number: $1134$
• Class group: $[3, 3, 126]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 468 x^{4} - 622 x^{3} + 74263 x^{2} - 49616 x + 3994081 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-594663237952=-\,2^{6}\cdot 7^{4}\cdot 157^{3}\)
Root discriminant:		$91.70$
Ramified primes:		$2, 7, 157$
This field is Galois and abelian over $\Q$.
Conductor:		\(4396=2^{2}\cdot 7\cdot 157\)
Dirichlet character group:		$\lbrace$$\chi_{4396}(1,·)$, $\chi_{4396}(627,·)$, $\chi_{4396}(1255,·)$, $\chi_{4396}(1257,·)$, $\chi_{4396}(3767,·)$, $\chi_{4396}(1885,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{253225349} a^{5} - \frac{36088966}{253225349} a^{4} - \frac{36289310}{253225349} a^{3} + \frac{80948568}{253225349} a^{2} - \frac{17619151}{253225349} a - \frac{3726940}{19478873}$

Class group and class number

$C_{3}\times C_{3}\times C_{126}$, which has order $1134$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1256}{253225349} a^{5} - \frac{403825}{253225349} a^{4} + \frac{1189460}{253225349} a^{3} - \frac{125188890}{253225349} a^{2} + \frac{154177056}{253225349} a - \frac{765783167}{19478873} \),  \( \frac{2526}{253225349} a^{5} + \frac{397524}{253225349} a^{4} + \frac{779278}{253225349} a^{3} + \frac{123226125}{253225349} a^{2} + \frac{61685998}{253225349} a + \frac{773200139}{19478873} \)
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

\(\Q(\sqrt{-157}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-157}) \) $\times$ \(\Q(\zeta_{7})^+\)

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	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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	$x^{3} - 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$157$	157.6.3.1	$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_157.2t1.1c1	$1$	$ 2^{2} \cdot 157 $	$x^{2} + 157$	$C_2$ (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_157.6t1.1c1	$1$	$ 2^{2} \cdot 7 \cdot 157 $	$x^{6} - 2 x^{5} + 468 x^{4} - 622 x^{3} + 74263 x^{2} - 49616 x + 3994081$	$C_6$ (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_157.6t1.1c2	$1$	$ 2^{2} \cdot 7 \cdot 157 $	$x^{6} - 2 x^{5} + 468 x^{4} - 622 x^{3} + 74263 x^{2} - 49616 x + 3994081$	$C_6$ (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 *.