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Properties

Label 14.0.38502899391...8827.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 127^{12}$
Root discriminant $110.11$
Ramified primes $3, 127$
Class number $2107$ (GRH)
Class group $[7, 301]$ (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![1256641, 1920273, 2970241, 1196220, 991629, 184596, 220112, 21346, 27699, -526, 2389, -8, 55, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 55*x^12 - 8*x^11 + 2389*x^10 - 526*x^9 + 27699*x^8 + 21346*x^7 + 220112*x^6 + 184596*x^5 + 991629*x^4 + 1196220*x^3 + 2970241*x^2 + 1920273*x + 1256641)
 
gp: K = bnfinit(x^14 - x^13 + 55*x^12 - 8*x^11 + 2389*x^10 - 526*x^9 + 27699*x^8 + 21346*x^7 + 220112*x^6 + 184596*x^5 + 991629*x^4 + 1196220*x^3 + 2970241*x^2 + 1920273*x + 1256641, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 55 x^{12} - 8 x^{11} + 2389 x^{10} - 526 x^{9} + 27699 x^{8} + 21346 x^{7} + 220112 x^{6} + 184596 x^{5} + 991629 x^{4} + 1196220 x^{3} + 2970241 x^{2} + 1920273 x + 1256641 \)

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:  \(-38502899391974811462791218827=-\,3^{7}\cdot 127^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.11$
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:  $3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(381=3\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{381}(32,·)$, $\chi_{381}(1,·)$, $\chi_{381}(2,·)$, $\chi_{381}(131,·)$, $\chi_{381}(4,·)$, $\chi_{381}(262,·)$, $\chi_{381}(8,·)$, $\chi_{381}(64,·)$, $\chi_{381}(128,·)$, $\chi_{381}(256,·)$, $\chi_{381}(143,·)$, $\chi_{381}(16,·)$, $\chi_{381}(286,·)$, $\chi_{381}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} + \frac{4}{19} a^{7} - \frac{9}{19} a^{6} - \frac{3}{19} a^{5} + \frac{4}{19} a^{4} - \frac{4}{19} a^{3} - \frac{2}{19} a^{2} - \frac{3}{19} a$, $\frac{1}{19} a^{9} - \frac{6}{19} a^{7} - \frac{5}{19} a^{6} - \frac{3}{19} a^{5} - \frac{1}{19} a^{4} - \frac{5}{19} a^{3} + \frac{5}{19} a^{2} - \frac{7}{19} a$, $\frac{1}{19} a^{10} + \frac{1}{19} a$, $\frac{1}{19} a^{11} + \frac{1}{19} a^{2}$, $\frac{1}{13357} a^{12} - \frac{275}{13357} a^{11} + \frac{329}{13357} a^{10} + \frac{26}{13357} a^{9} + \frac{87}{13357} a^{8} - \frac{207}{13357} a^{7} + \frac{3533}{13357} a^{6} - \frac{3759}{13357} a^{5} + \frac{4160}{13357} a^{4} + \frac{1309}{13357} a^{3} + \frac{289}{13357} a^{2} - \frac{298}{703} a + \frac{1}{37}$, $\frac{1}{47923643358268100783319591021239} a^{13} - \frac{46286990731427049764466583}{1295233604277516237387015973547} a^{12} + \frac{1051020318095302757364957411941}{47923643358268100783319591021239} a^{11} + \frac{1191909614936580248241659096186}{47923643358268100783319591021239} a^{10} - \frac{871944959408959510267290664359}{47923643358268100783319591021239} a^{9} - \frac{1052710104845313133166478901773}{47923643358268100783319591021239} a^{8} - \frac{5181681673175672281766636894624}{47923643358268100783319591021239} a^{7} - \frac{8262792604900607764568521898248}{47923643358268100783319591021239} a^{6} + \frac{680634458925386893696974457218}{2522297018856215830701031106381} a^{5} + \frac{14363022323576730464323539070694}{47923643358268100783319591021239} a^{4} + \frac{19550906327649752727514912236249}{47923643358268100783319591021239} a^{3} - \frac{12606870022278354558816851454065}{47923643358268100783319591021239} a^{2} + \frac{39652586258297349355171279923}{2522297018856215830701031106381} a + \frac{434362527023908342426633927}{2250041943671914211151678061}$

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

Class group and class number

$C_{7}\times C_{301}$, which has order $2107$ (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:  \( \frac{35474933512473195438492}{68170189698816644073000840713} a^{13} - \frac{25303175086855328037978}{68170189698816644073000840713} a^{12} + \frac{1953068755276894213436472}{68170189698816644073000840713} a^{11} + \frac{157868066277180523904677}{68170189698816644073000840713} a^{10} + \frac{85358900860091325513038985}{68170189698816644073000840713} a^{9} + \frac{147954809086642038373641}{68170189698816644073000840713} a^{8} + \frac{1003007784221497741027478840}{68170189698816644073000840713} a^{7} + \frac{794934126303847734225193443}{68170189698816644073000840713} a^{6} + \frac{8258135911967076194188213830}{68170189698816644073000840713} a^{5} + \frac{6802701663391554828509488074}{68170189698816644073000840713} a^{4} + \frac{36573570897297951304715061957}{68170189698816644073000840713} a^{3} + \frac{30513602778058112483407320627}{68170189698816644073000840713} a^{2} + \frac{110488640461098576831471185267}{68170189698816644073000840713} a + \frac{63830071491870838859470249}{60811944423565248950045353} \) (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:  \( 546287.2103473756 \) (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

\(\Q(\sqrt{-3}) \), 7.7.4195872914689.1

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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$127$127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$
127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$