Properties

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)

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sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 29*x^12 + 290*x^10 + 1247*x^8 + 2262*x^6 + 1566*x^4 + 377*x^2 + 29)
 
gp: K = bnfinit(x^14 + 29*x^12 + 290*x^10 + 1247*x^8 + 2262*x^6 + 1566*x^4 + 377*x^2 + 29, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 0, 377, 0, 1566, 0, 2262, 0, 1247, 0, 290, 0, 29, 0, 1]);
 

Normalized defining polynomial

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

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-168110140833113738264576=-\,2^{14}\cdot 29^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.60$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $14$
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 \(a\))

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

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 6020.98510015 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ 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.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.116.2t1.a.a$1$ $ 2^{2} \cdot 29 $ $x^{2} + 29$ $C_2$ (as 2T1) $1$ $-1$
* 1.29.7t1.a.a$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.a$1$ $ 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.a.b$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.b$1$ $ 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.a.c$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.c$1$ $ 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.a.d$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.d$1$ $ 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.a.e$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.e$1$ $ 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.a.f$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.f$1$ $ 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 *.