Global number field 12.0.222957405416602891976704.6

• Label: 12.0.22295740541...6704.6
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $2^{24}\cdot 7^{10}\cdot 19^{6}$
• Root discriminant: $88.24$
• Ramified primes: $2, 7, 19$
• Class number: $16120$ (GRH)
• Class group: $[2, 2, 4030]$ (GRH)
• Galois group: $C_6\times C_2$ (as 12T2)

Normalized defining polynomial

\( x^{12} + 128 x^{10} + 6493 x^{8} + 166076 x^{6} + 2261086 x^{4} + 16095776 x^{2} + 58232161 \)

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(222957405416602891976704=2^{24}\cdot 7^{10}\cdot 19^{6}\)
Root discriminant:		$88.24$
Ramified primes:		$2, 7, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(1064=2^{3}\cdot 7\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{1064}(1,·)$, $\chi_{1064}(227,·)$, $\chi_{1064}(837,·)$, $\chi_{1064}(1063,·)$, $\chi_{1064}(457,·)$, $\chi_{1064}(75,·)$, $\chi_{1064}(305,·)$, $\chi_{1064}(531,·)$, $\chi_{1064}(533,·)$, $\chi_{1064}(759,·)$, $\chi_{1064}(989,·)$, $\chi_{1064}(607,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7631} a^{7} + \frac{63}{7631} a^{5} + \frac{1134}{7631} a^{3} - \frac{2528}{7631} a$, $\frac{1}{14414959} a^{8} - \frac{2304499}{14414959} a^{6} + \frac{5289417}{14414959} a^{4} + \frac{6529608}{14414959} a^{2} - \frac{179}{1889}$, $\frac{1}{14414959} a^{9} + \frac{81}{14414959} a^{7} + \frac{6328367}{14414959} a^{5} - \frac{3599210}{14414959} a^{3} - \frac{3700753}{14414959} a$, $\frac{1}{14414959} a^{10} + \frac{5598319}{14414959} a^{6} + \frac{31291}{1108843} a^{4} + \frac{754482}{14414959} a^{2} - \frac{613}{1889}$, $\frac{1}{14414959} a^{11} - \frac{677}{14414959} a^{7} - \frac{490073}{1108843} a^{5} - \frac{5925022}{14414959} a^{3} - \frac{454281}{1108843} a$

Class group and class number

$C_{2}\times C_{2}\times C_{4030}$, which has order $16120$ (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:		\( 279.1500271937239 \) (assuming GRH)

Galois group

$C_2\times C_6$ (as 12T2):

An abelian group of order 12

The 12 conjugacy class representatives for $C_6\times C_2$

Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-266}) \), \(\Q(\sqrt{-133}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-133})\), 6.0.59022957056.3, 6.0.7377869632.2, 6.6.1229312.1

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

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$	R	${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$	R	${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$	2.12.24.318	$x^{12} + 60 x^{11} + 14 x^{10} + 36 x^{9} - 34 x^{8} - 32 x^{7} - 48 x^{6} - 32 x^{5} + 36 x^{4} - 16 x^{3} - 40 x^{2} - 48 x + 56$	$4$	$3$	$24$	$C_6\times C_2$	$[2, 3]^{3}$
$7$	7.6.5.2	$x^{6} - 7$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
	7.6.5.2	$x^{6} - 7$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
$19$	19.12.6.1	$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$