↑

Properties

Label 14.0.44208969295...7779.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,659^{13}$
Root discriminant $414.51$
Ramified prime $659$
Class number $17336$ (GRH)
Class group $[2, 2, 4334]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18994958023, 3458916720, 1793249698, 30774107, 229342015, 79585767, 25016954, 2644631, 53254, -14210, -4652, -760, 24, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 24*x^12 - 760*x^11 - 4652*x^10 - 14210*x^9 + 53254*x^8 + 2644631*x^7 + 25016954*x^6 + 79585767*x^5 + 229342015*x^4 + 30774107*x^3 + 1793249698*x^2 + 3458916720*x + 18994958023)
 
gp: K = bnfinit(x^14 - x^13 + 24*x^12 - 760*x^11 - 4652*x^10 - 14210*x^9 + 53254*x^8 + 2644631*x^7 + 25016954*x^6 + 79585767*x^5 + 229342015*x^4 + 30774107*x^3 + 1793249698*x^2 + 3458916720*x + 18994958023, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 24 x^{12} - 760 x^{11} - 4652 x^{10} - 14210 x^{9} + 53254 x^{8} + 2644631 x^{7} + 25016954 x^{6} + 79585767 x^{5} + 229342015 x^{4} + 30774107 x^{3} + 1793249698 x^{2} + 3458916720 x + 18994958023 \)

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:  \(-4420896929577970806479733313662687779=-\,659^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $414.51$
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:  $659$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(659\)
Dirichlet character group:    $\lbrace$$\chi_{659}(352,·)$, $\chi_{659}(1,·)$, $\chi_{659}(515,·)$, $\chi_{659}(389,·)$, $\chi_{659}(647,·)$, $\chi_{659}(12,·)$, $\chi_{659}(270,·)$, $\chi_{659}(144,·)$, $\chi_{659}(658,·)$, $\chi_{659}(307,·)$, $\chi_{659}(55,·)$, $\chi_{659}(249,·)$, $\chi_{659}(410,·)$, $\chi_{659}(604,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{1203920346508919899418718119764512142932078764350086564036283483} a^{13} + \frac{217770411684784519212737801779290970266130619723343435960601455}{1203920346508919899418718119764512142932078764350086564036283483} a^{12} + \frac{560499192761555945267032752502939532880888873758893243447824090}{1203920346508919899418718119764512142932078764350086564036283483} a^{11} - \frac{71163630165782203752532409018953836771999826033064139938405432}{1203920346508919899418718119764512142932078764350086564036283483} a^{10} + \frac{28947736482741670365427690294826297546913049202994349947985049}{1203920346508919899418718119764512142932078764350086564036283483} a^{9} - \frac{414897225126431159221293962360931469551038613832674948867524327}{1203920346508919899418718119764512142932078764350086564036283483} a^{8} - \frac{493306081088538076372581540304413058138907047885729032190967427}{1203920346508919899418718119764512142932078764350086564036283483} a^{7} + \frac{364805478768582552550286913297024956780197844557671467681732464}{1203920346508919899418718119764512142932078764350086564036283483} a^{6} + \frac{36940751655604016930996290190815696159508726764145445559063408}{1203920346508919899418718119764512142932078764350086564036283483} a^{5} - \frac{328939154800181793420064762455633787927637041940612314660379212}{1203920346508919899418718119764512142932078764350086564036283483} a^{4} + \frac{583660273041750747798781631115320147581797093712492068849901910}{1203920346508919899418718119764512142932078764350086564036283483} a^{3} - \frac{503022745831352850764195020487684060387019948320133848754390474}{1203920346508919899418718119764512142932078764350086564036283483} a^{2} + \frac{81459830963075093870851472651897797190765971034206772462099597}{1203920346508919899418718119764512142932078764350086564036283483} a - \frac{508315423965975544534968012979680355101189007636604496748149925}{1203920346508919899418718119764512142932078764350086564036283483}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4334}$, which has order $17336$ (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:  \( 89341256.28995533 \) (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

\(\Q(\sqrt{-659}) \), 7.7.81905390937410041.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
659Data not computed