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Properties

Label 14.0.10271873781...9056.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 113^{13}$
Root discriminant $228.02$
Ramified primes $2, 113$
Class number $131776$ (GRH)
Class group $[2, 2, 2, 16472]$ (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![72913024, 0, 76811072, 0, 28530240, 0, 4805664, 0, 396856, 0, 15368, 0, 226, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024)
 
gp: K = bnfinit(x^14 + 226*x^12 + 15368*x^10 + 396856*x^8 + 4805664*x^6 + 28530240*x^4 + 76811072*x^2 + 72913024, 1)
 

Normalized defining polynomial

\( x^{14} + 226 x^{12} + 15368 x^{10} + 396856 x^{8} + 4805664 x^{6} + 28530240 x^{4} + 76811072 x^{2} + 72913024 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1027187378113291173607353987629056=-\,2^{21}\cdot 113^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $228.02$
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:  $2, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(904=2^{3}\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{904}(897,·)$, $\chi_{904}(451,·)$, $\chi_{904}(1,·)$, $\chi_{904}(129,·)$, $\chi_{904}(795,·)$, $\chi_{904}(323,·)$, $\chi_{904}(459,·)$, $\chi_{904}(403,·)$, $\chi_{904}(369,·)$, $\chi_{904}(49,·)$, $\chi_{904}(83,·)$, $\chi_{904}(561,·)$, $\chi_{904}(593,·)$, $\chi_{904}(763,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{6435061622605376} a^{12} + \frac{37194421858289}{3217530811302688} a^{10} + \frac{23876330484727}{1608765405651344} a^{8} - \frac{1509416692538}{100547837853209} a^{6} - \frac{40402370965415}{402191351412836} a^{4} - \frac{39690005429053}{201095675706418} a^{2} + \frac{5594120455079}{100547837853209}$, $\frac{1}{456889375204981696} a^{13} + \frac{1847055503216051}{228444687602490848} a^{11} - \frac{1584889075166617}{114222343801245424} a^{9} + \frac{289568180019323}{57111171900622712} a^{7} - \frac{336881065764283}{7138896487577839} a^{5} - \frac{844072708254725}{14277792975155678} a^{3} + \frac{2619837904638513}{7138896487577839} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{16472}$, which has order $131776$ (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:  $6$
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:  \( 222748.97284811488 \) (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

\(\Q(\sqrt{-226}) \), 7.7.2081951752609.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$113$113.14.13.1$x^{14} - 113$$14$$1$$13$$C_{14}$$[\ ]_{14}$