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Properties

Label 14.0.17182641242...5243.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,43^{13}$
Root discriminant $32.87$
Ramified prime $43$
Class number $1$
Class group Trivial
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![79, 168, 235, -88, 163, 209, 83, -81, 143, 27, -17, 24, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 2*x^12 + 24*x^11 - 17*x^10 + 27*x^9 + 143*x^8 - 81*x^7 + 83*x^6 + 209*x^5 + 163*x^4 - 88*x^3 + 235*x^2 + 168*x + 79)
 
gp: K = bnfinit(x^14 - x^13 + 2*x^12 + 24*x^11 - 17*x^10 + 27*x^9 + 143*x^8 - 81*x^7 + 83*x^6 + 209*x^5 + 163*x^4 - 88*x^3 + 235*x^2 + 168*x + 79, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 2 x^{12} + 24 x^{11} - 17 x^{10} + 27 x^{9} + 143 x^{8} - 81 x^{7} + 83 x^{6} + 209 x^{5} + 163 x^{4} - 88 x^{3} + 235 x^{2} + 168 x + 79 \)

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:  \(-1718264124282290785243=-\,43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.87$
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:  $43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(43\)
Dirichlet character group:    $\lbrace$$\chi_{43}(32,·)$, $\chi_{43}(1,·)$, $\chi_{43}(2,·)$, $\chi_{43}(35,·)$, $\chi_{43}(4,·)$, $\chi_{43}(39,·)$, $\chi_{43}(8,·)$, $\chi_{43}(41,·)$, $\chi_{43}(42,·)$, $\chi_{43}(11,·)$, $\chi_{43}(16,·)$, $\chi_{43}(21,·)$, $\chi_{43}(22,·)$, $\chi_{43}(27,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{1744091421023249} a^{13} - \frac{82745671581626}{1744091421023249} a^{12} + \frac{261033923067570}{1744091421023249} a^{11} - \frac{29742838523662}{1744091421023249} a^{10} - \frac{503373242554238}{1744091421023249} a^{9} - \frac{604666142846306}{1744091421023249} a^{8} + \frac{818185152193630}{1744091421023249} a^{7} - \frac{489128085838790}{1744091421023249} a^{6} - \frac{324999217303861}{1744091421023249} a^{5} + \frac{84751421131921}{1744091421023249} a^{4} + \frac{312328981522648}{1744091421023249} a^{3} - \frac{584290881634991}{1744091421023249} a^{2} + \frac{215291569885636}{1744091421023249} a + \frac{9602647480904}{22077106595231}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35991.6418506 \)
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

\(\Q(\sqrt{-43}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ 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
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$