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Properties

Label 14.0.65061924746...3991.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,4871^{7}$
Root discriminant $69.79$
Ramified prime $4871$
Class number $13$ (GRH)
Class group $[13]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![496319, -318951, 322152, -182199, 80953, -37006, 13386, -6162, 3301, -690, -189, 128, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 16*x^12 + 128*x^11 - 189*x^10 - 690*x^9 + 3301*x^8 - 6162*x^7 + 13386*x^6 - 37006*x^5 + 80953*x^4 - 182199*x^3 + 322152*x^2 - 318951*x + 496319)
 
gp: K = bnfinit(x^14 - 4*x^13 - 16*x^12 + 128*x^11 - 189*x^10 - 690*x^9 + 3301*x^8 - 6162*x^7 + 13386*x^6 - 37006*x^5 + 80953*x^4 - 182199*x^3 + 322152*x^2 - 318951*x + 496319, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 16 x^{12} + 128 x^{11} - 189 x^{10} - 690 x^{9} + 3301 x^{8} - 6162 x^{7} + 13386 x^{6} - 37006 x^{5} + 80953 x^{4} - 182199 x^{3} + 322152 x^{2} - 318951 x + 496319 \)

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:  \(-65061924746822370167743991=-\,4871^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.79$
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:  $4871$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{16482325836065618583299053146840443} a^{13} + \frac{5716534110986971501009749624095465}{16482325836065618583299053146840443} a^{12} + \frac{1741458066905554858981702744102737}{16482325836065618583299053146840443} a^{11} + \frac{2623174126316273116389493020589565}{16482325836065618583299053146840443} a^{10} - \frac{278942750053093516606158426841570}{716622862437635590578219702036541} a^{9} + \frac{52739883189186930870828930316008}{716622862437635590578219702036541} a^{8} - \frac{6827370579199151122858839583885385}{16482325836065618583299053146840443} a^{7} - \frac{1720500554582471643748667570313840}{16482325836065618583299053146840443} a^{6} + \frac{7366677878203234231291867470148}{71975221991552919577725122911967} a^{5} - \frac{7427548403696713358057061352380699}{16482325836065618583299053146840443} a^{4} - \frac{3993442271577369775742132722401506}{16482325836065618583299053146840443} a^{3} + \frac{6499240956860967736107066591157014}{16482325836065618583299053146840443} a^{2} - \frac{1727331280271151316429904653273290}{16482325836065618583299053146840443} a + \frac{1623982222004845411512683885650704}{16482325836065618583299053146840443}$

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

Class group and class number

$C_{13}$, which has order $13$ (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:  \( 408655.333869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-4871}) \), 7.1.115572468311.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.115572468311.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
4871Data not computed