Global number field 14.0.168110140833113738264576.1

• Label: 14.0.16811014083...4576.1
• Degree: $14$
• Signature: $[0, 7]$
• Discriminant: $-\,2^{14}\cdot 29^{13}$
• Root discriminant: $45.60$
• Ramified primes: $2, 29$
• Class number: $48$ (GRH)
• Class group: $[2, 2, 2, 6]$ (GRH)
• Galois group: $C_{14}$ (as 14T1)

Normalized defining polynomial

\( x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29 \)

Invariants

Degree:		$14$
Signature:		$[0, 7]$
Discriminant:		\(-168110140833113738264576=-\,2^{14}\cdot 29^{13}\)
Root discriminant:		$45.60$
Ramified primes:		$2, 29$
This field is Galois and abelian over $\Q$.
Conductor:		\(116=2^{2}\cdot 29\)
Dirichlet character group:		$\lbrace$$\chi_{116}(1,·)$, $\chi_{116}(81,·)$, $\chi_{116}(67,·)$, $\chi_{116}(51,·)$, $\chi_{116}(65,·)$, $\chi_{116}(71,·)$, $\chi_{116}(45,·)$, $\chi_{116}(49,·)$, $\chi_{116}(115,·)$, $\chi_{116}(53,·)$, $\chi_{116}(25,·)$, $\chi_{116}(91,·)$, $\chi_{116}(35,·)$, $\chi_{116}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{5}{17} a^{8} + \frac{3}{17} a^{6} - \frac{4}{17} a^{4} + \frac{7}{17} a^{2} - \frac{6}{17}$, $\frac{1}{17} a^{11} - \frac{5}{17} a^{9} + \frac{3}{17} a^{7} - \frac{4}{17} a^{5} + \frac{7}{17} a^{3} - \frac{6}{17} a$, $\frac{1}{41123} a^{12} - \frac{75}{41123} a^{10} + \frac{15347}{41123} a^{8} + \frac{13794}{41123} a^{6} - \frac{12361}{41123} a^{4} - \frac{16731}{41123} a^{2} - \frac{18212}{41123}$, $\frac{1}{41123} a^{13} - \frac{75}{41123} a^{11} + \frac{15347}{41123} a^{9} + \frac{13794}{41123} a^{7} - \frac{12361}{41123} a^{5} - \frac{16731}{41123} a^{3} - \frac{18212}{41123} a$

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (assuming GRH)

Unit group

Rank:		$6$
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:		\( 6020.98510015 \) (assuming GRH)

Galois group

$C_{14}$ (as 14T1):

A cyclic group of order 14

The 14 conjugacy class representatives for $C_{14}$

Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-29}) \), 7.7.594823321.1

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	${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/7.14.0.1}{14} }$	${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$	${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/23.14.0.1}{14} }$	R	${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/37.14.0.1}{14} }$	${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$	${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$2$	2.14.14.15	$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$	$2$	$7$	$14$	$C_{14}$	$[2]^{7}$
$29$	29.14.13.1	$x^{14} - 29$	$14$	$1$	$13$	$C_{14}$	$[\ ]_{14}$

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_29.2t1.1c1	$1$	$ 2^{2} \cdot 29 $	$x^{2} + 29$	$C_2$ (as 2T1)	$1$	$-1$
*	1.29.7t1.1c1	$1$	$ 29 $	$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$	$C_7$ (as 7T1)	$0$	$1$
*	1.2e2_29.14t1.1c1	$1$	$ 2^{2} \cdot 29 $	$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$	$C_{14}$ (as 14T1)	$0$	$-1$
*	1.29.7t1.1c2	$1$	$ 29 $	$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$	$C_7$ (as 7T1)	$0$	$1$
*	1.2e2_29.14t1.1c2	$1$	$ 2^{2} \cdot 29 $	$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$	$C_{14}$ (as 14T1)	$0$	$-1$
*	1.29.7t1.1c3	$1$	$ 29 $	$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$	$C_7$ (as 7T1)	$0$	$1$
*	1.2e2_29.14t1.1c3	$1$	$ 2^{2} \cdot 29 $	$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$	$C_{14}$ (as 14T1)	$0$	$-1$
*	1.29.7t1.1c4	$1$	$ 29 $	$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$	$C_7$ (as 7T1)	$0$	$1$
*	1.2e2_29.14t1.1c4	$1$	$ 2^{2} \cdot 29 $	$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$	$C_{14}$ (as 14T1)	$0$	$-1$
*	1.29.7t1.1c5	$1$	$ 29 $	$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$	$C_7$ (as 7T1)	$0$	$1$
*	1.2e2_29.14t1.1c5	$1$	$ 2^{2} \cdot 29 $	$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$	$C_{14}$ (as 14T1)	$0$	$-1$
*	1.29.7t1.1c6	$1$	$ 29 $	$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$	$C_7$ (as 7T1)	$0$	$1$
*	1.2e2_29.14t1.1c6	$1$	$ 2^{2} \cdot 29 $	$x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$	$C_{14}$ (as 14T1)	$0$	$-1$

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 *.