Global number field 6.0.851324971.1

• Label: 6.0.851324971.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,7^{5}\cdot 37^{3}$
• Root discriminant: $30.79$
• Ramified primes: $7, 37$
• Class number: $28$
• Class group: $[28]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 64 x^{4} - 64 x^{3} + 1198 x^{2} - 1198 x + 6301 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-851324971=-\,7^{5}\cdot 37^{3}\)
Root discriminant:		$30.79$
Ramified primes:		$7, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(259=7\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{259}(1,·)$, $\chi_{259}(258,·)$, $\chi_{259}(149,·)$, $\chi_{259}(73,·)$, $\chi_{259}(186,·)$, $\chi_{259}(110,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{1261} a^{4} - \frac{128}{1261} a^{3} + \frac{36}{1261} a^{2} + \frac{327}{1261} a + \frac{162}{1261}$, $\frac{1}{1261} a^{5} + \frac{45}{1261} a^{3} - \frac{109}{1261} a^{2} + \frac{405}{1261} a + \frac{560}{1261}$

Class group and class number

$C_{28}$, which has order $28$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{1261} a^{5} + \frac{45}{1261} a^{3} - \frac{109}{1261} a^{2} + \frac{405}{1261} a - \frac{1962}{1261} \),  \( \frac{10}{1261} a^{4} - \frac{19}{1261} a^{3} + \frac{360}{1261} a^{2} - \frac{513}{1261} a + \frac{2881}{1261} \)
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

\(\Q(\sqrt{-259}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{-259}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.6.0.1}{6} }$	${\href{/LocalNumberField/3.6.0.1}{6} }$	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$	7.6.5.5	$x^{6} + 56$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
$37$	37.6.3.1	$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.7_37.2t1.1c1	$1$	$ 7 \cdot 37 $	$x^{2} - x + 65$	$C_2$ (as 2T1)	$1$	$-1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.7_37.6t1.8c1	$1$	$ 7 \cdot 37 $	$x^{6} - x^{5} + 64 x^{4} - 64 x^{3} + 1198 x^{2} - 1198 x + 6301$	$C_6$ (as 6T1)	$0$	$-1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.7_37.6t1.8c2	$1$	$ 7 \cdot 37 $	$x^{6} - x^{5} + 64 x^{4} - 64 x^{3} + 1198 x^{2} - 1198 x + 6301$	$C_6$ (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 *.