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Properties

Label 14.0.16431299768...8531.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,211^{13}$
Root discriminant $143.97$
Ramified prime $211$
Class number $843$ (GRH)
Class group $[843]$ (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![715937, 293289, -217661, 473241, 434837, -53486, -23157, 60342, 22837, -1653, -1089, 58, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 8*x^12 + 58*x^11 - 1089*x^10 - 1653*x^9 + 22837*x^8 + 60342*x^7 - 23157*x^6 - 53486*x^5 + 434837*x^4 + 473241*x^3 - 217661*x^2 + 293289*x + 715937)
 
gp: K = bnfinit(x^14 - x^13 + 8*x^12 + 58*x^11 - 1089*x^10 - 1653*x^9 + 22837*x^8 + 60342*x^7 - 23157*x^6 - 53486*x^5 + 434837*x^4 + 473241*x^3 - 217661*x^2 + 293289*x + 715937, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 8 x^{12} + 58 x^{11} - 1089 x^{10} - 1653 x^{9} + 22837 x^{8} + 60342 x^{7} - 23157 x^{6} - 53486 x^{5} + 434837 x^{4} + 473241 x^{3} - 217661 x^{2} + 293289 x + 715937 \)

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:  \(-1643129976812137607879885938531=-\,211^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.97$
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:  $211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(211\)
Dirichlet character group:    $\lbrace$$\chi_{211}(1,·)$, $\chi_{211}(67,·)$, $\chi_{211}(199,·)$, $\chi_{211}(40,·)$, $\chi_{211}(171,·)$, $\chi_{211}(12,·)$, $\chi_{211}(144,·)$, $\chi_{211}(210,·)$, $\chi_{211}(148,·)$, $\chi_{211}(88,·)$, $\chi_{211}(153,·)$, $\chi_{211}(58,·)$, $\chi_{211}(123,·)$, $\chi_{211}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} + \frac{3}{19} a^{7} + \frac{8}{19} a^{6} + \frac{4}{19} a^{5} - \frac{1}{19} a^{4} + \frac{7}{19} a^{3} - \frac{8}{19} a^{2} + \frac{3}{19} a - \frac{2}{19}$, $\frac{1}{19} a^{9} - \frac{1}{19} a^{7} - \frac{1}{19} a^{6} + \frac{6}{19} a^{5} - \frac{9}{19} a^{4} + \frac{9}{19} a^{3} + \frac{8}{19} a^{2} + \frac{8}{19} a + \frac{6}{19}$, $\frac{1}{19} a^{10} + \frac{2}{19} a^{7} - \frac{5}{19} a^{6} - \frac{5}{19} a^{5} + \frac{8}{19} a^{4} - \frac{4}{19} a^{3} + \frac{9}{19} a - \frac{2}{19}$, $\frac{1}{361} a^{11} - \frac{9}{361} a^{10} + \frac{8}{361} a^{9} + \frac{8}{361} a^{8} - \frac{89}{361} a^{7} - \frac{15}{361} a^{6} - \frac{179}{361} a^{5} + \frac{112}{361} a^{4} + \frac{74}{361} a^{3} - \frac{127}{361} a^{2} - \frac{115}{361} a + \frac{92}{361}$, $\frac{1}{60914779} a^{12} - \frac{4005}{3206041} a^{11} + \frac{522864}{60914779} a^{10} - \frac{1140718}{60914779} a^{9} - \frac{1492182}{60914779} a^{8} + \frac{1285}{6859} a^{7} - \frac{29546207}{60914779} a^{6} + \frac{21134975}{60914779} a^{5} + \frac{5133628}{60914779} a^{4} - \frac{3850}{60914779} a^{3} + \frac{25704393}{60914779} a^{2} + \frac{29750663}{60914779} a + \frac{89559}{569297}$, $\frac{1}{44618670312913621594677331} a^{13} + \frac{274137764829395863}{44618670312913621594677331} a^{12} - \frac{10414751691013600536527}{44618670312913621594677331} a^{11} - \frac{566028329251178504598215}{44618670312913621594677331} a^{10} + \frac{7190828353416866190709}{2348351069100716926035649} a^{9} + \frac{56843754348439867204708}{2348351069100716926035649} a^{8} + \frac{12891722822888503159285711}{44618670312913621594677331} a^{7} + \frac{5998243466674080392005854}{44618670312913621594677331} a^{6} - \frac{10712673188382857590166722}{44618670312913621594677331} a^{5} + \frac{1594305983395314882844732}{44618670312913621594677331} a^{4} + \frac{17873620607335910436327878}{44618670312913621594677331} a^{3} - \frac{10183711233703727448686043}{44618670312913621594677331} a^{2} + \frac{11214597295333115782099605}{44618670312913621594677331} a - \frac{113443063743872363472247}{416996918812276837333433}$

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

Class group and class number

$C_{843}$, which has order $843$ (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:  \( 2943138.71663 \) (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

\(\Q(\sqrt{-211}) \), 7.7.88245939632761.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
211Data not computed