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Properties

Label 14.14.8380141964...9952.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{21}\cdot 43^{12}$
Root discriminant $71.07$
Ramified primes $2, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 78, -1359, -1996, 20214, -26074, 86, 13320, -3225, -2164, 690, 130, -49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7)
 
gp: K = bnfinit(x^14 - 2*x^13 - 49*x^12 + 130*x^11 + 690*x^10 - 2164*x^9 - 3225*x^8 + 13320*x^7 + 86*x^6 - 26074*x^5 + 20214*x^4 - 1996*x^3 - 1359*x^2 + 78*x + 7, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 49 x^{12} + 130 x^{11} + 690 x^{10} - 2164 x^{9} - 3225 x^{8} + 13320 x^{7} + 86 x^{6} - 26074 x^{5} + 20214 x^{4} - 1996 x^{3} - 1359 x^{2} + 78 x + 7 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(83801419645740806624509952=2^{21}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.07$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(133,·)$, $\chi_{344}(97,·)$, $\chi_{344}(145,·)$, $\chi_{344}(41,·)$, $\chi_{344}(193,·)$, $\chi_{344}(269,·)$, $\chi_{344}(173,·)$, $\chi_{344}(305,·)$, $\chi_{344}(213,·)$, $\chi_{344}(121,·)$, $\chi_{344}(293,·)$, $\chi_{344}(317,·)$, $\chi_{344}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} + \frac{3}{49} a^{9} - \frac{3}{49} a^{8} + \frac{2}{49} a^{7} + \frac{1}{49} a^{6} - \frac{15}{49} a^{5} - \frac{2}{7} a^{4} + \frac{17}{49} a^{3} + \frac{18}{49} a^{2} + \frac{11}{49} a - \frac{3}{7}$, $\frac{1}{49} a^{11} + \frac{2}{49} a^{9} - \frac{3}{49} a^{8} + \frac{2}{49} a^{7} + \frac{3}{49} a^{6} + \frac{10}{49} a^{5} - \frac{18}{49} a^{4} - \frac{19}{49} a^{3} - \frac{8}{49} a^{2} + \frac{16}{49} a + \frac{2}{7}$, $\frac{1}{343} a^{12} - \frac{2}{343} a^{11} + \frac{2}{343} a^{10} - \frac{2}{49} a^{9} + \frac{8}{343} a^{8} - \frac{22}{343} a^{7} + \frac{4}{343} a^{6} + \frac{158}{343} a^{5} + \frac{17}{343} a^{4} - \frac{110}{343} a^{3} - \frac{164}{343} a^{2} + \frac{3}{343} a + \frac{17}{49}$, $\frac{1}{26431843081} a^{13} + \frac{4344747}{26431843081} a^{12} - \frac{47361633}{26431843081} a^{11} + \frac{110836676}{26431843081} a^{10} + \frac{720011363}{26431843081} a^{9} + \frac{1762804521}{26431843081} a^{8} + \frac{852064088}{26431843081} a^{7} + \frac{261596610}{26431843081} a^{6} + \frac{5208466355}{26431843081} a^{5} - \frac{364560010}{26431843081} a^{4} + \frac{7921916222}{26431843081} a^{3} - \frac{7352728186}{26431843081} a^{2} - \frac{9948901142}{26431843081} a + \frac{249168117}{3775977583}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1198577302.5055304 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{2}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$