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Properties

Label 14.14.4898011103...7553.1
Degree $14$
Signature $[14, 0]$
Discriminant $113^{13}$
Root discriminant $80.62$
Ramified prime $113$
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![-1583, 874, 8945, -3381, -18257, 3600, 16495, -856, -6351, -162, 908, 31, -52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 52*x^12 + 31*x^11 + 908*x^10 - 162*x^9 - 6351*x^8 - 856*x^7 + 16495*x^6 + 3600*x^5 - 18257*x^4 - 3381*x^3 + 8945*x^2 + 874*x - 1583)
 
gp: K = bnfinit(x^14 - x^13 - 52*x^12 + 31*x^11 + 908*x^10 - 162*x^9 - 6351*x^8 - 856*x^7 + 16495*x^6 + 3600*x^5 - 18257*x^4 - 3381*x^3 + 8945*x^2 + 874*x - 1583, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 52 x^{12} + 31 x^{11} + 908 x^{10} - 162 x^{9} - 6351 x^{8} - 856 x^{7} + 16495 x^{6} + 3600 x^{5} - 18257 x^{4} - 3381 x^{3} + 8945 x^{2} + 874 x - 1583 \)

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:  \(489801110321660601428677553=113^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.62$
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:  $113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(113\)
Dirichlet character group:    $\lbrace$$\chi_{113}(64,·)$, $\chi_{113}(1,·)$, $\chi_{113}(4,·)$, $\chi_{113}(16,·)$, $\chi_{113}(7,·)$, $\chi_{113}(106,·)$, $\chi_{113}(109,·)$, $\chi_{113}(112,·)$, $\chi_{113}(49,·)$, $\chi_{113}(83,·)$, $\chi_{113}(97,·)$, $\chi_{113}(28,·)$, $\chi_{113}(30,·)$, $\chi_{113}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{35578803801412210091} a^{13} + \frac{6903031322650652525}{35578803801412210091} a^{12} - \frac{3254387418069573411}{35578803801412210091} a^{11} - \frac{15355461326495579806}{35578803801412210091} a^{10} - \frac{13869289227140863170}{35578803801412210091} a^{9} + \frac{15735208656282060120}{35578803801412210091} a^{8} + \frac{15258045001869129938}{35578803801412210091} a^{7} + \frac{13485182429524347922}{35578803801412210091} a^{6} + \frac{7832743953553772518}{35578803801412210091} a^{5} + \frac{11334368163651726455}{35578803801412210091} a^{4} - \frac{12027358933835694225}{35578803801412210091} a^{3} + \frac{17547397941017992219}{35578803801412210091} a^{2} + \frac{6633733564656019584}{35578803801412210091} a + \frac{17250476101097264885}{35578803801412210091}$

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:  \( 392671339.436 \) (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

\(\Q(\sqrt{113}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$113$113.14.13.1$x^{14} - 113$$14$$1$$13$$C_{14}$$[\ ]_{14}$