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Properties

Label 14.0.31977829362...2211.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,11^{7}\cdot 71^{12}$
Root discriminant $128.08$
Ramified primes $11, 71$
Class number $25781$ (GRH)
Class group $[25781]$ (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![6578125, -5606250, 3964475, -1936270, 750422, -101414, -84021, 38489, 3573, -4561, 380, 211, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 30*x^12 + 211*x^11 + 380*x^10 - 4561*x^9 + 3573*x^8 + 38489*x^7 - 84021*x^6 - 101414*x^5 + 750422*x^4 - 1936270*x^3 + 3964475*x^2 - 5606250*x + 6578125)
 
gp: K = bnfinit(x^14 - 5*x^13 - 30*x^12 + 211*x^11 + 380*x^10 - 4561*x^9 + 3573*x^8 + 38489*x^7 - 84021*x^6 - 101414*x^5 + 750422*x^4 - 1936270*x^3 + 3964475*x^2 - 5606250*x + 6578125, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 30 x^{12} + 211 x^{11} + 380 x^{10} - 4561 x^{9} + 3573 x^{8} + 38489 x^{7} - 84021 x^{6} - 101414 x^{5} + 750422 x^{4} - 1936270 x^{3} + 3964475 x^{2} - 5606250 x + 6578125 \)

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:  \(-319778293622616136151658322211=-\,11^{7}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.08$
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:  $11, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(781=11\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{781}(32,·)$, $\chi_{781}(1,·)$, $\chi_{781}(329,·)$, $\chi_{781}(747,·)$, $\chi_{781}(45,·)$, $\chi_{781}(527,·)$, $\chi_{781}(529,·)$, $\chi_{781}(243,·)$, $\chi_{781}(659,·)$, $\chi_{781}(758,·)$, $\chi_{781}(375,·)$, $\chi_{781}(474,·)$, $\chi_{781}(463,·)$, $\chi_{781}(285,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{8} - \frac{1}{25} a^{7} - \frac{2}{25} a^{6} - \frac{1}{25} a^{5} + \frac{8}{25} a^{4} - \frac{4}{25} a^{3} - \frac{8}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{10625} a^{12} - \frac{127}{10625} a^{11} - \frac{11}{10625} a^{10} - \frac{22}{10625} a^{9} + \frac{314}{10625} a^{8} + \frac{506}{10625} a^{7} + \frac{791}{10625} a^{6} - \frac{363}{10625} a^{5} + \frac{34}{125} a^{4} + \frac{1381}{10625} a^{3} + \frac{193}{2125} a^{2} - \frac{78}{425} a - \frac{3}{17}$, $\frac{1}{395096767277274207755099375} a^{13} - \frac{7070690104995536835161}{395096767277274207755099375} a^{12} - \frac{1639657912034592778784168}{395096767277274207755099375} a^{11} - \frac{6138182279587974213040823}{395096767277274207755099375} a^{10} - \frac{1751780396112595954904338}{395096767277274207755099375} a^{9} + \frac{1579120833201016870922481}{79019353455454841551019875} a^{8} + \frac{16336137829330572778810812}{395096767277274207755099375} a^{7} - \frac{35364980357199097719293232}{395096767277274207755099375} a^{6} - \frac{2653996084361862448636118}{395096767277274207755099375} a^{5} - \frac{124748607774358763919894704}{395096767277274207755099375} a^{4} + \frac{3830221980464935330857633}{23240986310427894573829375} a^{3} + \frac{27661690757121406759393278}{79019353455454841551019875} a^{2} + \frac{96861170362010188747601}{929639452417115782953175} a - \frac{101956509869058259865694}{632154827643638732408159}$

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

Class group and class number

$C_{25781}$, which has order $25781$ (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:  \( 315114.6966253571 \) (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

\(\Q(\sqrt{-11}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ R ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$11$11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$71$71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$