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Properties

Label 14.0.70507022062...2539.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,17^{7}\cdot 43^{13}$
Root discriminant $135.52$
Ramified primes $17, 43$
Class number $161124$ (GRH)
Class group $[161124]$ (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![104926787, 114689596, 105212807, 65360600, 27679091, 11236797, 3316243, 881591, 232859, 27547, 9615, 196, 174, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 174*x^12 + 196*x^11 + 9615*x^10 + 27547*x^9 + 232859*x^8 + 881591*x^7 + 3316243*x^6 + 11236797*x^5 + 27679091*x^4 + 65360600*x^3 + 105212807*x^2 + 114689596*x + 104926787)
 
gp: K = bnfinit(x^14 - x^13 + 174*x^12 + 196*x^11 + 9615*x^10 + 27547*x^9 + 232859*x^8 + 881591*x^7 + 3316243*x^6 + 11236797*x^5 + 27679091*x^4 + 65360600*x^3 + 105212807*x^2 + 114689596*x + 104926787, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 174 x^{12} + 196 x^{11} + 9615 x^{10} + 27547 x^{9} + 232859 x^{8} + 881591 x^{7} + 3316243 x^{6} + 11236797 x^{5} + 27679091 x^{4} + 65360600 x^{3} + 105212807 x^{2} + 114689596 x + 104926787 \)

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:  \(-705070220621502278216740602539=-\,17^{7}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.52$
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:  $17, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(731=17\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{731}(256,·)$, $\chi_{731}(1,·)$, $\chi_{731}(35,·)$, $\chi_{731}(613,·)$, $\chi_{731}(237,·)$, $\chi_{731}(494,·)$, $\chi_{731}(118,·)$, $\chi_{731}(696,·)$, $\chi_{731}(730,·)$, $\chi_{731}(475,·)$, $\chi_{731}(188,·)$, $\chi_{731}(477,·)$, $\chi_{731}(254,·)$, $\chi_{731}(543,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{259} a^{11} + \frac{10}{259} a^{9} + \frac{13}{259} a^{8} - \frac{4}{259} a^{7} - \frac{2}{37} a^{6} - \frac{36}{259} a^{5} - \frac{12}{37} a^{4} - \frac{129}{259} a^{3} - \frac{6}{259} a^{2} + \frac{123}{259} a + \frac{8}{37}$, $\frac{1}{1813} a^{12} - \frac{3}{1813} a^{11} + \frac{47}{1813} a^{10} - \frac{17}{1813} a^{9} + \frac{31}{1813} a^{8} - \frac{39}{1813} a^{7} + \frac{117}{1813} a^{6} + \frac{505}{1813} a^{5} - \frac{765}{1813} a^{4} + \frac{233}{1813} a^{3} + \frac{30}{1813} a^{2} - \frac{13}{259} a - \frac{14}{37}$, $\frac{1}{31689101806067097714656344105940999506457} a^{13} + \frac{7769708437191808203313522803476245588}{31689101806067097714656344105940999506457} a^{12} - \frac{55183785469827840056902108351076249395}{31689101806067097714656344105940999506457} a^{11} + \frac{80447631694062939461780501230805819850}{4527014543723871102093763443705857072351} a^{10} - \frac{1290373728355066271878474721238830616022}{31689101806067097714656344105940999506457} a^{9} - \frac{394964484147958876437612310916329279208}{31689101806067097714656344105940999506457} a^{8} + \frac{1232960867213302978141056471842938800336}{31689101806067097714656344105940999506457} a^{7} + \frac{741729310317977859120349922919491513059}{31689101806067097714656344105940999506457} a^{6} + \frac{744059968333682371423125186245380951}{92388051912732063308035988647058307599} a^{5} + \frac{7045808671830322380479308667689381211785}{31689101806067097714656344105940999506457} a^{4} + \frac{2001405864698256935357231460356200108799}{31689101806067097714656344105940999506457} a^{3} - \frac{4455129799053378820074925330328426740141}{31689101806067097714656344105940999506457} a^{2} + \frac{2038001997087358677777472170817695552040}{4527014543723871102093763443705857072351} a + \frac{74847754145922641084950380523655981929}{646716363389124443156251920529408153193}$

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

Class group and class number

$C_{161124}$, which has order $161124$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-731}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$17$17.14.7.1$x^{14} - 9826 x^{8} + 24137569 x^{2} - 3693048057$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$