↑

Properties

Label 14.0.28606260651...8503.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 239^{12}$
Root discriminant $289.19$
Ramified primes $7, 239$
Class number $105112$ (GRH)
Class group $[2, 2, 26278]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![438678011, -355150475, 47158292, 79245585, -21789837, -11390256, 4463486, 673209, -337442, -21350, 12356, 565, -181, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 181*x^12 + 565*x^11 + 12356*x^10 - 21350*x^9 - 337442*x^8 + 673209*x^7 + 4463486*x^6 - 11390256*x^5 - 21789837*x^4 + 79245585*x^3 + 47158292*x^2 - 355150475*x + 438678011)
 
gp: K = bnfinit(x^14 - 5*x^13 - 181*x^12 + 565*x^11 + 12356*x^10 - 21350*x^9 - 337442*x^8 + 673209*x^7 + 4463486*x^6 - 11390256*x^5 - 21789837*x^4 + 79245585*x^3 + 47158292*x^2 - 355150475*x + 438678011, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 181 x^{12} + 565 x^{11} + 12356 x^{10} - 21350 x^{9} - 337442 x^{8} + 673209 x^{7} + 4463486 x^{6} - 11390256 x^{5} - 21789837 x^{4} + 79245585 x^{3} + 47158292 x^{2} - 355150475 x + 438678011 \)

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:  \(-28606260651310274757079509174248503=-\,7^{7}\cdot 239^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $289.19$
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:  $7, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1673=7\cdot 239\)
Dirichlet character group:    $\lbrace$$\chi_{1673}(1056,·)$, $\chi_{1673}(1,·)$, $\chi_{1673}(1219,·)$, $\chi_{1673}(741,·)$, $\chi_{1673}(1478,·)$, $\chi_{1673}(1000,·)$, $\chi_{1673}(1196,·)$, $\chi_{1673}(337,·)$, $\chi_{1673}(1205,·)$, $\chi_{1673}(918,·)$, $\chi_{1673}(727,·)$, $\chi_{1673}(440,·)$, $\chi_{1673}(1532,·)$, $\chi_{1673}(1534,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23} a^{9} - \frac{8}{23} a^{8} - \frac{10}{23} a^{7} - \frac{1}{23} a^{6} + \frac{2}{23} a^{5} - \frac{5}{23} a^{4} - \frac{2}{23} a^{3} + \frac{7}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{23} a^{10} - \frac{5}{23} a^{8} + \frac{11}{23} a^{7} - \frac{6}{23} a^{6} + \frac{11}{23} a^{5} + \frac{4}{23} a^{4} - \frac{9}{23} a^{3} + \frac{1}{23} a^{2} - \frac{3}{23} a$, $\frac{1}{23} a^{11} - \frac{6}{23} a^{8} - \frac{10}{23} a^{7} + \frac{6}{23} a^{6} - \frac{9}{23} a^{5} - \frac{11}{23} a^{4} - \frac{9}{23} a^{3} + \frac{9}{23} a^{2} + \frac{1}{23} a$, $\frac{1}{10323343} a^{12} + \frac{221288}{10323343} a^{11} - \frac{1149}{448841} a^{10} + \frac{137619}{10323343} a^{9} + \frac{104666}{10323343} a^{8} - \frac{422392}{10323343} a^{7} + \frac{5028748}{10323343} a^{6} - \frac{2763175}{10323343} a^{5} - \frac{3531725}{10323343} a^{4} + \frac{3216850}{10323343} a^{3} + \frac{340627}{10323343} a^{2} - \frac{412483}{10323343} a + \frac{214016}{448841}$, $\frac{1}{35203662682407731687729534859743155831927} a^{13} + \frac{1571370371299645006411352870322556}{35203662682407731687729534859743155831927} a^{12} - \frac{328600998098186482885685753065118324}{1530594029669901377727371080858398079649} a^{11} + \frac{568473073128897617458227596432083844633}{35203662682407731687729534859743155831927} a^{10} + \frac{652822132168416987225235089292815415451}{35203662682407731687729534859743155831927} a^{9} - \frac{2111256207307092600496202217820601180257}{35203662682407731687729534859743155831927} a^{8} + \frac{10077761831568072344448195040372621338478}{35203662682407731687729534859743155831927} a^{7} + \frac{13010341067209153399710128705313996548992}{35203662682407731687729534859743155831927} a^{6} - \frac{12676517351630051337650935075165280511966}{35203662682407731687729534859743155831927} a^{5} + \frac{13979195505536368838834543768492169412666}{35203662682407731687729534859743155831927} a^{4} - \frac{16389865340579019946605614733865406011047}{35203662682407731687729534859743155831927} a^{3} - \frac{11072969943773071677449048750060592854574}{35203662682407731687729534859743155831927} a^{2} + \frac{304738506561906264704848214318190070367}{1530594029669901377727371080858398079649} a - \frac{25649828659481908210450825960935220253}{66547566507387016422929177428626003463}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{26278}$, which has order $105112$ (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:  \( 3022802.0673343516 \) (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

\(\Q(\sqrt{-7}) \), 7.7.186374892382561.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
239Data not computed