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Properties

Label 14.0.19837600798...7763.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,883^{13}$
Root discriminant $543.92$
Ramified prime $883$
Class number $3489$ (GRH)
Class group $[3489]$ (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![132193055349, -41024712654, 18337865967, -9840332223, 2734738830, -654977697, 98559596, -6931854, 1057143, -49474, -3350, -766, 32, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 32*x^12 - 766*x^11 - 3350*x^10 - 49474*x^9 + 1057143*x^8 - 6931854*x^7 + 98559596*x^6 - 654977697*x^5 + 2734738830*x^4 - 9840332223*x^3 + 18337865967*x^2 - 41024712654*x + 132193055349)
 
gp: K = bnfinit(x^14 - x^13 + 32*x^12 - 766*x^11 - 3350*x^10 - 49474*x^9 + 1057143*x^8 - 6931854*x^7 + 98559596*x^6 - 654977697*x^5 + 2734738830*x^4 - 9840332223*x^3 + 18337865967*x^2 - 41024712654*x + 132193055349, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 32 x^{12} - 766 x^{11} - 3350 x^{10} - 49474 x^{9} + 1057143 x^{8} - 6931854 x^{7} + 98559596 x^{6} - 654977697 x^{5} + 2734738830 x^{4} - 9840332223 x^{3} + 18337865967 x^{2} - 41024712654 x + 132193055349 \)

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:  \(-198376007989432085363129684509608357763=-\,883^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $543.92$
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:  $883$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(883\)
Dirichlet character group:    $\lbrace$$\chi_{883}(1,·)$, $\chi_{883}(707,·)$, $\chi_{883}(134,·)$, $\chi_{883}(199,·)$, $\chi_{883}(296,·)$, $\chi_{883}(684,·)$, $\chi_{883}(587,·)$, $\chi_{883}(812,·)$, $\chi_{883}(749,·)$, $\chi_{883}(176,·)$, $\chi_{883}(626,·)$, $\chi_{883}(71,·)$, $\chi_{883}(882,·)$, $\chi_{883}(257,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{10} - \frac{1}{27} a^{8} - \frac{2}{27} a^{7} + \frac{2}{27} a^{6} + \frac{4}{27} a^{5} + \frac{13}{27} a^{4} - \frac{2}{27} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{11} - \frac{1}{81} a^{10} - \frac{4}{81} a^{9} - \frac{1}{81} a^{8} - \frac{11}{81} a^{7} + \frac{8}{81} a^{6} + \frac{1}{27} a^{5} - \frac{2}{9} a^{4} + \frac{38}{81} a^{3} + \frac{1}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{729} a^{12} - \frac{1}{729} a^{11} + \frac{5}{729} a^{10} - \frac{37}{729} a^{9} - \frac{2}{729} a^{8} + \frac{71}{729} a^{7} + \frac{13}{243} a^{6} + \frac{2}{81} a^{5} + \frac{146}{729} a^{4} + \frac{115}{243} a^{3} + \frac{34}{81} a^{2} - \frac{1}{9} a$, $\frac{1}{1568511970661082679829691394788030682189570483997112489129} a^{13} - \frac{412491726632676219078549014935548877578339127146365362}{1568511970661082679829691394788030682189570483997112489129} a^{12} - \frac{5768905920442546442825539246125287083025692338756724885}{1568511970661082679829691394788030682189570483997112489129} a^{11} + \frac{19559096499734771929242122947593306260338727800733625869}{1568511970661082679829691394788030682189570483997112489129} a^{10} + \frac{54155020805077185033922277723794764757802142808007563804}{1568511970661082679829691394788030682189570483997112489129} a^{9} - \frac{75257145530879223007770864432519255717048361586372265373}{1568511970661082679829691394788030682189570483997112489129} a^{8} - \frac{33875505914380289706133908943003091895669785419825573364}{522837323553694226609897131596010227396523494665704163043} a^{7} - \frac{8648524523350407114269358421635033182771510988881547565}{174279107851231408869965710532003409132174498221901387681} a^{6} + \frac{195081185468665769089330110476925988104726900565628509019}{1568511970661082679829691394788030682189570483997112489129} a^{5} - \frac{183244497193027314472742875780770701248150007630055587960}{522837323553694226609897131596010227396523494665704163043} a^{4} + \frac{24295593931865898163913190699705496602595761719349194860}{174279107851231408869965710532003409132174498221901387681} a^{3} - \frac{3153198395384106366180611515749791962447979433758047003}{19364345316803489874440634503555934348019388691322376409} a^{2} + \frac{773681045648554860809728110992195865592662682814953888}{2151593924089276652715626055950659372002154299035819601} a - \frac{899645174822680550284920306820116700538117249193991}{3367126641767256107536191010877401208141086539962159}$

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

Class group and class number

$C_{3489}$, which has order $3489$ (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:  \( 8256963011.914384 \) (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

\(\Q(\sqrt{-883}) \), 7.7.473984589097059769.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
883Data not computed