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Properties

Label 14.0.26885624202...4544.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 71^{12}$
Root discriminant $77.24$
Ramified primes $2, 71$
Class number $3277$ (GRH)
Class group $[3277]$ (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![18769, 0, 127429, 0, 185798, 0, 80418, 0, 15247, 0, 1402, 0, 61, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 61*x^12 + 1402*x^10 + 15247*x^8 + 80418*x^6 + 185798*x^4 + 127429*x^2 + 18769)
 
gp: K = bnfinit(x^14 + 61*x^12 + 1402*x^10 + 15247*x^8 + 80418*x^6 + 185798*x^4 + 127429*x^2 + 18769, 1)
 

Normalized defining polynomial

\( x^{14} + 61 x^{12} + 1402 x^{10} + 15247 x^{8} + 80418 x^{6} + 185798 x^{4} + 127429 x^{2} + 18769 \)

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:  \(-268856242022659049623404544=-\,2^{14}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.24$
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:  $2, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(284=2^{2}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{284}(1,·)$, $\chi_{284}(91,·)$, $\chi_{284}(261,·)$, $\chi_{284}(103,·)$, $\chi_{284}(233,·)$, $\chi_{284}(45,·)$, $\chi_{284}(143,·)$, $\chi_{284}(243,·)$, $\chi_{284}(245,·)$, $\chi_{284}(119,·)$, $\chi_{284}(179,·)$, $\chi_{284}(187,·)$, $\chi_{284}(37,·)$, $\chi_{284}(101,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{1}{5}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{8} - \frac{2}{25} a^{4} + \frac{1}{25}$, $\frac{1}{25} a^{9} - \frac{2}{25} a^{5} + \frac{1}{25} a$, $\frac{1}{2125} a^{10} + \frac{31}{2125} a^{8} + \frac{198}{2125} a^{6} - \frac{37}{2125} a^{4} - \frac{349}{2125} a^{2} - \frac{144}{2125}$, $\frac{1}{2125} a^{11} + \frac{31}{2125} a^{9} + \frac{198}{2125} a^{7} - \frac{37}{2125} a^{5} - \frac{349}{2125} a^{3} - \frac{144}{2125} a$, $\frac{1}{3006875} a^{12} - \frac{132}{601375} a^{10} + \frac{34367}{3006875} a^{8} + \frac{23034}{601375} a^{6} + \frac{140138}{3006875} a^{4} + \frac{137623}{601375} a^{2} - \frac{1054881}{3006875}$, $\frac{1}{411941875} a^{13} - \frac{15414}{82388375} a^{11} + \frac{311707}{411941875} a^{9} - \frac{3483902}{82388375} a^{7} - \frac{34197667}{411941875} a^{5} + \frac{25196141}{82388375} a^{3} + \frac{92577084}{411941875} a$

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

Class group and class number

$C_{3277}$, which has order $3277$ (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:  \( -\frac{27589}{411941875} a^{13} - \frac{332421}{82388375} a^{11} - \frac{37496783}{411941875} a^{9} - \frac{79139288}{82388375} a^{7} - \frac{1972918042}{411941875} a^{5} - \frac{799173166}{82388375} a^{3} - \frac{1645974336}{411941875} a \) (order $4$)
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:  \( 315114.696625 \) (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

\(\Q(\sqrt{-1}) \), 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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$71$71.14.12.1$x^{14} + 546629 x^{7} + 98234829011$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$