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Properties

Label 14.0.22764194375...6875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{7}\cdot 29^{12}$
Root discriminant $106.05$
Ramified primes $5, 7, 29$
Class number $42448$ (GRH)
Class group $[2, 2, 2, 5306]$ (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![33177421, -10152668, 14758999, -3877996, 2903596, -650470, 328657, -61247, 24153, -3759, 1260, -169, 48, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 48*x^12 - 169*x^11 + 1260*x^10 - 3759*x^9 + 24153*x^8 - 61247*x^7 + 328657*x^6 - 650470*x^5 + 2903596*x^4 - 3877996*x^3 + 14758999*x^2 - 10152668*x + 33177421)
 
gp: K = bnfinit(x^14 - 5*x^13 + 48*x^12 - 169*x^11 + 1260*x^10 - 3759*x^9 + 24153*x^8 - 61247*x^7 + 328657*x^6 - 650470*x^5 + 2903596*x^4 - 3877996*x^3 + 14758999*x^2 - 10152668*x + 33177421, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 48 x^{12} - 169 x^{11} + 1260 x^{10} - 3759 x^{9} + 24153 x^{8} - 61247 x^{7} + 328657 x^{6} - 650470 x^{5} + 2903596 x^{4} - 3877996 x^{3} + 14758999 x^{2} - 10152668 x + 33177421 \)

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:  \(-22764194375420436752520546875=-\,5^{7}\cdot 7^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.05$
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:  $5, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1015=5\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1015}(384,·)$, $\chi_{1015}(1,·)$, $\chi_{1015}(36,·)$, $\chi_{1015}(806,·)$, $\chi_{1015}(489,·)$, $\chi_{1015}(139,·)$, $\chi_{1015}(141,·)$, $\chi_{1015}(944,·)$, $\chi_{1015}(596,·)$, $\chi_{1015}(981,·)$, $\chi_{1015}(281,·)$, $\chi_{1015}(314,·)$, $\chi_{1015}(349,·)$, $\chi_{1015}(629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{4}{17} a^{9} + \frac{7}{17} a^{8} + \frac{3}{17} a^{7} + \frac{3}{17} a^{6} + \frac{1}{17} a^{5} + \frac{2}{17} a^{4} - \frac{8}{17} a^{3} + \frac{5}{17} a^{2} - \frac{2}{17} a$, $\frac{1}{17} a^{11} + \frac{8}{17} a^{9} - \frac{3}{17} a^{8} - \frac{2}{17} a^{7} - \frac{4}{17} a^{6} + \frac{6}{17} a^{5} + \frac{7}{17} a^{3} + \frac{1}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{17} a^{12} - \frac{5}{17} a^{9} - \frac{7}{17} a^{8} + \frac{6}{17} a^{7} - \frac{1}{17} a^{6} - \frac{8}{17} a^{5} + \frac{8}{17} a^{4} - \frac{3}{17} a^{3} + \frac{3}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{916285702686575360946415314612847} a^{13} + \frac{11066191335911672042244026010877}{916285702686575360946415314612847} a^{12} + \frac{18954367305149895558846519867923}{916285702686575360946415314612847} a^{11} + \frac{8364904915771871262025807541963}{916285702686575360946415314612847} a^{10} + \frac{80695968051886127399830247782463}{916285702686575360946415314612847} a^{9} - \frac{137447439491646327824910401289037}{916285702686575360946415314612847} a^{8} + \frac{297637492833666525165092143568990}{916285702686575360946415314612847} a^{7} - \frac{28262680300761606811614742200099}{916285702686575360946415314612847} a^{6} + \frac{328879118701897009495350393027731}{916285702686575360946415314612847} a^{5} + \frac{359191029478574712914799856612729}{916285702686575360946415314612847} a^{4} + \frac{115639819701585992310630361677370}{916285702686575360946415314612847} a^{3} + \frac{318750550717682737679797095497428}{916285702686575360946415314612847} a^{2} - \frac{237684925227505330356648696678666}{916285702686575360946415314612847} a + \frac{11624392692914305048343819251}{311555832263371425007281643867}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{5306}$, which has order $42448$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-35}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$