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Properties

Label 14.0.11650874745...1111.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,71^{13}$
Root discriminant $52.36$
Ramified prime $71$
Class number $49$ (GRH)
Class group $[49]$ (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![625, 1750, 4050, 755, 3202, 4490, 1560, -749, -250, 156, 44, -11, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 3*x^12 - 11*x^11 + 44*x^10 + 156*x^9 - 250*x^8 - 749*x^7 + 1560*x^6 + 4490*x^5 + 3202*x^4 + 755*x^3 + 4050*x^2 + 1750*x + 625)
 
gp: K = bnfinit(x^14 - x^13 + 3*x^12 - 11*x^11 + 44*x^10 + 156*x^9 - 250*x^8 - 749*x^7 + 1560*x^6 + 4490*x^5 + 3202*x^4 + 755*x^3 + 4050*x^2 + 1750*x + 625, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 3 x^{12} - 11 x^{11} + 44 x^{10} + 156 x^{9} - 250 x^{8} - 749 x^{7} + 1560 x^{6} + 4490 x^{5} + 3202 x^{4} + 755 x^{3} + 4050 x^{2} + 1750 x + 625 \)

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:  \(-1165087474585497590531111=-\,71^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.36$
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:  $71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(71\)
Dirichlet character group:    $\lbrace$$\chi_{71}(32,·)$, $\chi_{71}(1,·)$, $\chi_{71}(34,·)$, $\chi_{71}(37,·)$, $\chi_{71}(70,·)$, $\chi_{71}(39,·)$, $\chi_{71}(41,·)$, $\chi_{71}(45,·)$, $\chi_{71}(48,·)$, $\chi_{71}(51,·)$, $\chi_{71}(20,·)$, $\chi_{71}(23,·)$, $\chi_{71}(26,·)$, $\chi_{71}(30,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} - \frac{1}{25} a^{6} - \frac{1}{25} a^{5} - \frac{4}{25} a^{4} + \frac{4}{25} a^{3} - \frac{4}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{625} a^{12} - \frac{1}{125} a^{11} + \frac{3}{625} a^{10} + \frac{2}{625} a^{9} + \frac{1}{625} a^{8} - \frac{38}{625} a^{7} + \frac{32}{625} a^{6} + \frac{8}{625} a^{5} - \frac{12}{625} a^{4} + \frac{28}{625} a^{3} + \frac{36}{125} a^{2} - \frac{3}{25} a - \frac{1}{5}$, $\frac{1}{4106020289175625} a^{13} + \frac{338608684971}{821204057835125} a^{12} + \frac{3135028373353}{4106020289175625} a^{11} - \frac{9490134612893}{4106020289175625} a^{10} + \frac{78031410280046}{4106020289175625} a^{9} + \frac{318104873948997}{4106020289175625} a^{8} + \frac{7310256529206}{241530605245625} a^{7} - \frac{209048708414097}{4106020289175625} a^{6} - \frac{276667560518457}{4106020289175625} a^{5} - \frac{663192587381592}{4106020289175625} a^{4} - \frac{19727965082058}{821204057835125} a^{3} - \frac{13523529657442}{164240811567025} a^{2} + \frac{1307071334437}{32848162313405} a - \frac{2428089719090}{6569632462681}$

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

Class group and class number

$C_{49}$, which has order $49$ (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:  \( 315114.696625 \) (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

\(\Q(\sqrt{-71}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$71$71.14.13.11$x^{14} + 9088$$14$$1$$13$$C_{14}$$[\ ]_{14}$