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Properties

Label 14.0.17030925399...5627.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 211^{12}$
Root discriminant $170.14$
Ramified primes $3, 211$
Class number $5887$ (GRH)
Class group $[29, 203]$ (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![21743569, -24476087, 28130213, -11528880, 7192317, -1305976, 1215344, -102522, 111679, 8698, 6725, 228, 91, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 91*x^12 + 228*x^11 + 6725*x^10 + 8698*x^9 + 111679*x^8 - 102522*x^7 + 1215344*x^6 - 1305976*x^5 + 7192317*x^4 - 11528880*x^3 + 28130213*x^2 - 24476087*x + 21743569)
 
gp: K = bnfinit(x^14 - x^13 + 91*x^12 + 228*x^11 + 6725*x^10 + 8698*x^9 + 111679*x^8 - 102522*x^7 + 1215344*x^6 - 1305976*x^5 + 7192317*x^4 - 11528880*x^3 + 28130213*x^2 - 24476087*x + 21743569, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 91 x^{12} + 228 x^{11} + 6725 x^{10} + 8698 x^{9} + 111679 x^{8} - 102522 x^{7} + 1215344 x^{6} - 1305976 x^{5} + 7192317 x^{4} - 11528880 x^{3} + 28130213 x^{2} - 24476087 x + 21743569 \)

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:  \(-17030925399469881272195784585627=-\,3^{7}\cdot 211^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $170.14$
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:  $3, 211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(633=3\cdot 211\)
Dirichlet character group:    $\lbrace$$\chi_{633}(1,·)$, $\chi_{633}(355,·)$, $\chi_{633}(359,·)$, $\chi_{633}(199,·)$, $\chi_{633}(269,·)$, $\chi_{633}(334,·)$, $\chi_{633}(593,·)$, $\chi_{633}(148,·)$, $\chi_{633}(545,·)$, $\chi_{633}(566,·)$, $\chi_{633}(212,·)$, $\chi_{633}(58,·)$, $\chi_{633}(410,·)$, $\chi_{633}(382,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} - \frac{3}{19} a^{8} + \frac{4}{19} a^{7} + \frac{1}{19} a^{5} + \frac{7}{19} a^{4} - \frac{9}{19} a^{3} + \frac{4}{19} a^{2} + \frac{3}{19} a - \frac{8}{19}$, $\frac{1}{19} a^{10} - \frac{5}{19} a^{8} - \frac{7}{19} a^{7} + \frac{1}{19} a^{6} - \frac{9}{19} a^{5} - \frac{7}{19} a^{4} - \frac{4}{19} a^{3} - \frac{4}{19} a^{2} + \frac{1}{19} a - \frac{5}{19}$, $\frac{1}{19} a^{11} - \frac{3}{19} a^{8} + \frac{2}{19} a^{7} - \frac{9}{19} a^{6} - \frac{2}{19} a^{5} - \frac{7}{19} a^{4} + \frac{8}{19} a^{3} + \frac{2}{19} a^{2} - \frac{9}{19} a - \frac{2}{19}$, $\frac{1}{69388171} a^{12} - \frac{219325}{69388171} a^{11} - \frac{977965}{69388171} a^{10} + \frac{606946}{69388171} a^{9} - \frac{17610900}{69388171} a^{8} - \frac{27712938}{69388171} a^{7} - \frac{24170606}{69388171} a^{6} + \frac{34010869}{69388171} a^{5} - \frac{5556055}{69388171} a^{4} + \frac{13653399}{69388171} a^{3} - \frac{5345008}{69388171} a^{2} - \frac{12088201}{69388171} a - \frac{5982177}{69388171}$, $\frac{1}{46955808285802910919222505146990615949} a^{13} + \frac{113386930799887453961976310945}{46955808285802910919222505146990615949} a^{12} - \frac{333260248814170972640369791363854032}{46955808285802910919222505146990615949} a^{11} - \frac{1097452646732781877906943258042578227}{46955808285802910919222505146990615949} a^{10} + \frac{798756681558497730200333894646301112}{46955808285802910919222505146990615949} a^{9} - \frac{10031560517005284044267247703875739557}{46955808285802910919222505146990615949} a^{8} - \frac{983897040919763418794489155395646865}{46955808285802910919222505146990615949} a^{7} + \frac{6674402545059493277146959401197776248}{46955808285802910919222505146990615949} a^{6} + \frac{7177685452389081708175477044447815867}{46955808285802910919222505146990615949} a^{5} + \frac{1745727116249880616439378628491968231}{46955808285802910919222505146990615949} a^{4} + \frac{44775502726242667237440994683807165}{2041556881991430909531413267260461563} a^{3} - \frac{22743273332831553004337807085440324727}{46955808285802910919222505146990615949} a^{2} + \frac{524618243794656932279878850826786192}{2041556881991430909531413267260461563} a + \frac{2284549734185663109815601404183646}{10069870959854795393356745688824923}$

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:  \( \frac{31993851055684142285271528}{784309213212228547649409630142321} a^{13} - \frac{42040344115927508344408370}{784309213212228547649409630142321} a^{12} + \frac{2938541043200525724682668992}{784309213212228547649409630142321} a^{11} + \frac{6515119069452645596048131735}{784309213212228547649409630142321} a^{10} + \frac{214114835999124985705040735123}{784309213212228547649409630142321} a^{9} + \frac{228033937717734967775625825611}{784309213212228547649409630142321} a^{8} + \frac{3623363037471507076993492127932}{784309213212228547649409630142321} a^{7} - \frac{3310181147883456235723728300859}{784309213212228547649409630142321} a^{6} + \frac{42122633133230728073594548691234}{784309213212228547649409630142321} a^{5} - \frac{42672980441024604833679486440082}{784309213212228547649409630142321} a^{4} + \frac{239405087284394175164376908232427}{784309213212228547649409630142321} a^{3} - \frac{254854610753148938715693839860515}{784309213212228547649409630142321} a^{2} + \frac{947730491330493086351134545958921}{784309213212228547649409630142321} a - \frac{9176854698999162430479248428}{168198415872234301447439337367} \) (order $6$)
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:  \( 2943138.716633699 \) (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

\(\Q(\sqrt{-3}) \), 7.7.88245939632761.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
211Data not computed