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Properties

Label 14.0.19960072708...7223.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 281^{12}$
Root discriminant $332.23$
Ramified primes $7, 281$
Class number $103936$ (GRH)
Class group $[2, 2, 2, 4, 4, 812]$ (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![40539809, 29680369, 16150610, 11488539, 9043905, 2570346, -490648, -156315, 205552, 103162, 15080, -287, -217, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 217*x^12 - 287*x^11 + 15080*x^10 + 103162*x^9 + 205552*x^8 - 156315*x^7 - 490648*x^6 + 2570346*x^5 + 9043905*x^4 + 11488539*x^3 + 16150610*x^2 + 29680369*x + 40539809)
 
gp: K = bnfinit(x^14 - 5*x^13 - 217*x^12 - 287*x^11 + 15080*x^10 + 103162*x^9 + 205552*x^8 - 156315*x^7 - 490648*x^6 + 2570346*x^5 + 9043905*x^4 + 11488539*x^3 + 16150610*x^2 + 29680369*x + 40539809, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 217 x^{12} - 287 x^{11} + 15080 x^{10} + 103162 x^{9} + 205552 x^{8} - 156315 x^{7} - 490648 x^{6} + 2570346 x^{5} + 9043905 x^{4} + 11488539 x^{3} + 16150610 x^{2} + 29680369 x + 40539809 \)

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:  \(-199600727086494002686044072453147223=-\,7^{7}\cdot 281^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $332.23$
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:  $7, 281$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1967=7\cdot 281\)
Dirichlet character group:    $\lbrace$$\chi_{1967}(1,·)$, $\chi_{1967}(1765,·)$, $\chi_{1967}(902,·)$, $\chi_{1967}(743,·)$, $\chi_{1967}(1289,·)$, $\chi_{1967}(811,·)$, $\chi_{1967}(1233,·)$, $\chi_{1967}(1203,·)$, $\chi_{1967}(181,·)$, $\chi_{1967}(727,·)$, $\chi_{1967}(1464,·)$, $\chi_{1967}(1373,·)$, $\chi_{1967}(1406,·)$, $\chi_{1967}(671,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{12175079} a^{12} - \frac{59912}{1739297} a^{11} + \frac{138246}{12175079} a^{10} + \frac{2728552}{12175079} a^{9} + \frac{4235611}{12175079} a^{8} - \frac{2744873}{12175079} a^{7} + \frac{1552984}{12175079} a^{6} - \frac{4984802}{12175079} a^{5} + \frac{3112761}{12175079} a^{4} - \frac{1492524}{12175079} a^{3} + \frac{5037727}{12175079} a^{2} + \frac{4440640}{12175079} a - \frac{5833082}{12175079}$, $\frac{1}{814616481086952227388755188583768572753} a^{13} + \frac{390363122694376438421379030248}{814616481086952227388755188583768572753} a^{12} - \frac{25633054139070274673888435281131677031}{814616481086952227388755188583768572753} a^{11} - \frac{56401514387578568836338231077871928910}{814616481086952227388755188583768572753} a^{10} + \frac{378044552332454359515886876182157032535}{814616481086952227388755188583768572753} a^{9} - \frac{25953515754864437024613426208458274980}{116373783012421746769822169797681224679} a^{8} + \frac{74275966003028218300872475381992411470}{814616481086952227388755188583768572753} a^{7} + \frac{304966529352821026803387428755836545707}{814616481086952227388755188583768572753} a^{6} + \frac{277794330777592988474595143307856142432}{814616481086952227388755188583768572753} a^{5} - \frac{240172771820938725228370141254611771658}{814616481086952227388755188583768572753} a^{4} + \frac{22297883983807621528510472883923589680}{116373783012421746769822169797681224679} a^{3} + \frac{395897094122405342504939234750441890956}{814616481086952227388755188583768572753} a^{2} - \frac{84448339991678269578871086776558311003}{814616481086952227388755188583768572753} a - \frac{156549769469293329175201495473017674577}{814616481086952227388755188583768572753}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{812}$, which has order $103936$ (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:  \( 12176100.831221418 \) (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

\(\Q(\sqrt{-7}) \), 7.7.492309163417681.1

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
281Data not computed