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Properties

Label 14.0.32918715676...4443.6
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{13}$
Root discriminant $923.70$
Ramified primes $7, 43$
Class number $7685608$ (GRH)
Class group $[2, 2, 1921402]$ (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![65986926254491, 6892330333119, 4793924304218, 540194248076, 128672845938, 12020663158, 1745978192, 93883663, 13901986, 721798, 64414, -1505, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 1505*x^11 + 64414*x^10 + 721798*x^9 + 13901986*x^8 + 93883663*x^7 + 1745978192*x^6 + 12020663158*x^5 + 128672845938*x^4 + 540194248076*x^3 + 4793924304218*x^2 + 6892330333119*x + 65986926254491)
 
gp: K = bnfinit(x^14 - 1505*x^11 + 64414*x^10 + 721798*x^9 + 13901986*x^8 + 93883663*x^7 + 1745978192*x^6 + 12020663158*x^5 + 128672845938*x^4 + 540194248076*x^3 + 4793924304218*x^2 + 6892330333119*x + 65986926254491, 1)
 

Normalized defining polynomial

\( x^{14} - 1505 x^{11} + 64414 x^{10} + 721798 x^{9} + 13901986 x^{8} + 93883663 x^{7} + 1745978192 x^{6} + 12020663158 x^{5} + 128672845938 x^{4} + 540194248076 x^{3} + 4793924304218 x^{2} + 6892330333119 x + 65986926254491 \)

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:  \(-329187156767051876908796215815371233484443=-\,7^{24}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $923.70$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(2080,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(729,·)$, $\chi_{2107}(1387,·)$, $\chi_{2107}(1870,·)$, $\chi_{2107}(911,·)$, $\chi_{2107}(78,·)$, $\chi_{2107}(1527,·)$, $\chi_{2107}(1464,·)$, $\chi_{2107}(505,·)$, $\chi_{2107}(1114,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(477,·)$, $\chi_{2107}(414,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} + \frac{2}{11} a^{6} + \frac{1}{11} a^{4} - \frac{5}{11} a^{3} - \frac{2}{11} a^{2} + \frac{2}{11} a$, $\frac{1}{1199} a^{10} + \frac{4}{109} a^{9} + \frac{6}{1199} a^{8} + \frac{355}{1199} a^{7} - \frac{177}{1199} a^{6} - \frac{32}{1199} a^{5} - \frac{26}{109} a^{4} + \frac{325}{1199} a^{3} + \frac{520}{1199} a^{2} + \frac{593}{1199} a - \frac{11}{109}$, $\frac{1}{20383} a^{11} - \frac{8}{20383} a^{10} - \frac{39}{1853} a^{9} + \frac{370}{20383} a^{8} - \frac{4358}{20383} a^{7} + \frac{6883}{20383} a^{6} + \frac{6174}{20383} a^{5} + \frac{24}{1853} a^{4} - \frac{5262}{20383} a^{3} - \frac{9770}{20383} a^{2} + \frac{8719}{20383} a + \frac{8}{109}$, $\frac{1}{835703} a^{12} - \frac{12}{835703} a^{11} + \frac{18}{75973} a^{10} - \frac{18059}{835703} a^{9} - \frac{280218}{835703} a^{8} - \frac{201768}{835703} a^{7} + \frac{4890}{835703} a^{6} - \frac{6}{1853} a^{5} + \frac{42166}{835703} a^{4} + \frac{171299}{835703} a^{3} + \frac{246019}{835703} a^{2} - \frac{303153}{835703} a + \frac{2090}{4469}$, $\frac{1}{226585828525987787094583020510561008993521293356431273139164133261} a^{13} + \frac{104885467862261620230066498326056446480734930639617124550880}{226585828525987787094583020510561008993521293356431273139164133261} a^{12} - \frac{5022443300448358531197152546353865898214625640884479299739659}{226585828525987787094583020510561008993521293356431273139164133261} a^{11} - \frac{10205520343539737418271090874137012481198360630671822572694200}{226585828525987787094583020510561008993521293356431273139164133261} a^{10} - \frac{3758926475420905390753024388540583351185365206345640675263249345}{226585828525987787094583020510561008993521293356431273139164133261} a^{9} + \frac{106332641258965110458939539093425565101261001376041591406875906378}{226585828525987787094583020510561008993521293356431273139164133261} a^{8} + \frac{110905868146511370079763546931178526189674957891391085767914124335}{226585828525987787094583020510561008993521293356431273139164133261} a^{7} - \frac{100403627582042510915351481357147668589982190802936855185186683246}{226585828525987787094583020510561008993521293356431273139164133261} a^{6} - \frac{6575855832570655429947396147882090590828126607058321363360833938}{226585828525987787094583020510561008993521293356431273139164133261} a^{5} + \frac{79100447825709361513944583736965231229683250970552705018296242603}{226585828525987787094583020510561008993521293356431273139164133261} a^{4} - \frac{69104145996609096487215139028972531343166111623077344454160720378}{226585828525987787094583020510561008993521293356431273139164133261} a^{3} + \frac{18865838937948931102347258525686966740151537915870081011780860}{465268641737141246600786489754745398343986228657969760039351403} a^{2} + \frac{2579714485474939768850903921859666908453247724718336698437309}{17179909661535202600241339033327849646942246823597791579283049} a - \frac{36815594453573092320937082730294093909895553080225297488534610}{110153538418078651966253291448984447736276759045421134243638373}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1921402}$, which has order $7685608$ (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:  \( 769768169.7365845 \) (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

\(\Q(\sqrt{-43}) \), 7.7.87495801462998035849.6

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$43$43.14.13.9$x^{14} + 31347$$14$$1$$13$$C_{14}$$[\ ]_{14}$