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Properties

Label 14.0.14720064844...9799.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 197^{13}$
Root discriminant $233.96$
Ramified primes $3, 197$
Class number $34672$ (GRH)
Class group $[2, 2, 2, 4334]$ (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![1175495059, -886029565, 191487834, -7131374, 6623518, -1979796, -88208, 151316, 45656, 16405, 3759, 223, 106, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 106*x^12 + 223*x^11 + 3759*x^10 + 16405*x^9 + 45656*x^8 + 151316*x^7 - 88208*x^6 - 1979796*x^5 + 6623518*x^4 - 7131374*x^3 + 191487834*x^2 - 886029565*x + 1175495059)
 
gp: K = bnfinit(x^14 - x^13 + 106*x^12 + 223*x^11 + 3759*x^10 + 16405*x^9 + 45656*x^8 + 151316*x^7 - 88208*x^6 - 1979796*x^5 + 6623518*x^4 - 7131374*x^3 + 191487834*x^2 - 886029565*x + 1175495059, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 106 x^{12} + 223 x^{11} + 3759 x^{10} + 16405 x^{9} + 45656 x^{8} + 151316 x^{7} - 88208 x^{6} - 1979796 x^{5} + 6623518 x^{4} - 7131374 x^{3} + 191487834 x^{2} - 886029565 x + 1175495059 \)

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:  \(-1472006484410430025621015411349799=-\,3^{7}\cdot 197^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $233.96$
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:  $3, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(591=3\cdot 197\)
Dirichlet character group:    $\lbrace$$\chi_{591}(1,·)$, $\chi_{591}(290,·)$, $\chi_{591}(388,·)$, $\chi_{591}(230,·)$, $\chi_{591}(161,·)$, $\chi_{591}(361,·)$, $\chi_{591}(203,·)$, $\chi_{591}(301,·)$, $\chi_{591}(430,·)$, $\chi_{591}(178,·)$, $\chi_{591}(83,·)$, $\chi_{591}(590,·)$, $\chi_{591}(508,·)$, $\chi_{591}(413,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2147} a^{11} - \frac{142}{2147} a^{10} - \frac{1054}{2147} a^{9} + \frac{287}{2147} a^{8} - \frac{806}{2147} a^{7} + \frac{667}{2147} a^{6} - \frac{630}{2147} a^{5} + \frac{961}{2147} a^{4} - \frac{316}{2147} a^{3} - \frac{29}{113} a^{2} - \frac{564}{2147} a + \frac{36}{113}$, $\frac{1}{2147} a^{12} + \frac{252}{2147} a^{10} + \frac{909}{2147} a^{9} - \frac{845}{2147} a^{8} + \frac{6}{2147} a^{7} - \frac{384}{2147} a^{6} - \frac{472}{2147} a^{5} + \frac{885}{2147} a^{4} - \frac{336}{2147} a^{3} + \frac{633}{2147} a^{2} + \frac{35}{2147} a + \frac{27}{113}$, $\frac{1}{877926006816011973623058339589158712500401413471} a^{13} - \frac{151276146087683258298744499691801111422217302}{877926006816011973623058339589158712500401413471} a^{12} - \frac{123458090693913350371063426796777033639106510}{877926006816011973623058339589158712500401413471} a^{11} - \frac{384164646638767731006092114013226208609769809032}{877926006816011973623058339589158712500401413471} a^{10} - \frac{8859595531926479178598161656908509993926117079}{46206631937684840717003070504692563815810600709} a^{9} - \frac{35719123102146236086833477468197585592914549890}{877926006816011973623058339589158712500401413471} a^{8} + \frac{261017778668053571939981532662329623809192390145}{877926006816011973623058339589158712500401413471} a^{7} - \frac{159650086106516461409383621519499415890443581699}{877926006816011973623058339589158712500401413471} a^{6} + \frac{380817795527750939750196434385801744411492331496}{877926006816011973623058339589158712500401413471} a^{5} + \frac{209372160687293090876063147211634344823899222019}{877926006816011973623058339589158712500401413471} a^{4} - \frac{333620648614681256513459072251927754889313005789}{877926006816011973623058339589158712500401413471} a^{3} + \frac{141893782913415806336074224906158761357876995438}{877926006816011973623058339589158712500401413471} a^{2} + \frac{12285306214605006858122032683977324639924367107}{46206631937684840717003070504692563815810600709} a - \frac{622247083263252374316196661310735503224311002}{2431927996720254774579108973931187569253189511}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4334}$, which has order $34672$ (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:  \( 1553055.199048291 \) (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

\(\Q(\sqrt{-591}) \), 7.7.58451728309129.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$197$197.14.13.11$x^{14} + 25216$$14$$1$$13$$C_{14}$$[\ ]_{14}$