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Properties

Label 14.0.16929349193...4719.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7759^{7}$
Root discriminant $88.09$
Ramified prime $7759$
Class number $5887$ (GRH)
Class group $[29, 203]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4773375, -4392150, -851090, 1284789, 1093209, 233126, 79816, 4867, -1828, -2372, 223, -131, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 44*x^12 - 131*x^11 + 223*x^10 - 2372*x^9 - 1828*x^8 + 4867*x^7 + 79816*x^6 + 233126*x^5 + 1093209*x^4 + 1284789*x^3 - 851090*x^2 - 4392150*x + 4773375)
 
gp: K = bnfinit(x^14 - 4*x^13 + 44*x^12 - 131*x^11 + 223*x^10 - 2372*x^9 - 1828*x^8 + 4867*x^7 + 79816*x^6 + 233126*x^5 + 1093209*x^4 + 1284789*x^3 - 851090*x^2 - 4392150*x + 4773375, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 44 x^{12} - 131 x^{11} + 223 x^{10} - 2372 x^{9} - 1828 x^{8} + 4867 x^{7} + 79816 x^{6} + 233126 x^{5} + 1093209 x^{4} + 1284789 x^{3} - 851090 x^{2} - 4392150 x + 4773375 \)

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:  \(-1692934919397644000811644719=-\,7759^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.09$
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:  $7759$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{15} a^{8} - \frac{1}{15} a^{7} + \frac{1}{15} a^{6} - \frac{1}{15} a^{5} + \frac{4}{15} a^{4} - \frac{4}{15} a^{3} + \frac{4}{15} a^{2} - \frac{4}{15} a$, $\frac{1}{75} a^{9} + \frac{2}{75} a^{8} + \frac{4}{75} a^{7} - \frac{7}{75} a^{6} + \frac{4}{75} a^{5} + \frac{23}{75} a^{4} - \frac{29}{75} a^{3} + \frac{32}{75} a^{2} - \frac{2}{5} a$, $\frac{1}{75} a^{10} + \frac{1}{25} a^{6} - \frac{4}{75} a^{2}$, $\frac{1}{75} a^{11} + \frac{1}{25} a^{7} - \frac{4}{75} a^{3}$, $\frac{1}{88875} a^{12} - \frac{7}{5925} a^{11} + \frac{98}{29625} a^{10} + \frac{86}{17775} a^{9} + \frac{1718}{88875} a^{8} + \frac{1436}{17775} a^{7} + \frac{3467}{88875} a^{6} + \frac{929}{17775} a^{5} + \frac{26581}{88875} a^{4} + \frac{278}{711} a^{3} - \frac{22036}{88875} a^{2} - \frac{89}{395} a - \frac{164}{395}$, $\frac{1}{9491972496215048118683035417125} a^{13} - \frac{1626975831588496223055089}{1898394499243009623736607083425} a^{12} + \frac{6908625552600524330895243778}{3163990832071682706227678472375} a^{11} + \frac{1857268230119320340702377397}{1898394499243009623736607083425} a^{10} - \frac{22548891308789681365248020147}{9491972496215048118683035417125} a^{9} + \frac{468841817476287073237683859}{24030310117000121819450722575} a^{8} + \frac{14970099050914237111671721552}{9491972496215048118683035417125} a^{7} - \frac{85210371165172834254937480772}{1898394499243009623736607083425} a^{6} - \frac{99604190512180584076221298506}{1054663610690560902075892824125} a^{5} - \frac{68719918726577354137329284623}{210932722138112180415178564825} a^{4} + \frac{3225664378677471667738862135464}{9491972496215048118683035417125} a^{3} - \frac{99772606101261828223020543668}{379678899848601924747321416685} a^{2} + \frac{30414093372137946750590199433}{126559633282867308249107138895} a + \frac{25930995074620850959559012}{8437308885524487216607142593}$

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

Class group and class number

$C_{29}\times C_{203}$, which has order $5887$ (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:  \( 255747.453243 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-7759}) \), 7.1.467107946479.1 x7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 7 sibling: 7.1.467107946479.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
7759Data not computed