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Properties

Label 14.0.15720442004...6832.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 7^{7}\cdot 71^{13}$
Root discriminant $277.08$
Ramified primes $2, 7, 71$
Class number $673032$ (GRH)
Class group $[2, 336516]$ (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![16898278817, 0, 9681218561, 0, 1668229206, 0, 125807598, 0, 4554011, 0, 76538, 0, 497, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 497*x^12 + 76538*x^10 + 4554011*x^8 + 125807598*x^6 + 1668229206*x^4 + 9681218561*x^2 + 16898278817)
 
gp: K = bnfinit(x^14 + 497*x^12 + 76538*x^10 + 4554011*x^8 + 125807598*x^6 + 1668229206*x^4 + 9681218561*x^2 + 16898278817, 1)
 

Normalized defining polynomial

\( x^{14} + 497 x^{12} + 76538 x^{10} + 4554011 x^{8} + 125807598 x^{6} + 1668229206 x^{4} + 9681218561 x^{2} + 16898278817 \)

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:  \(-15720442004808735820984528834936832=-\,2^{14}\cdot 7^{7}\cdot 71^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $277.08$
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:  $2, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1988=2^{2}\cdot 7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{1988}(897,·)$, $\chi_{1988}(1091,·)$, $\chi_{1988}(1,·)$, $\chi_{1988}(1035,·)$, $\chi_{1988}(813,·)$, $\chi_{1988}(335,·)$, $\chi_{1988}(531,·)$, $\chi_{1988}(1681,·)$, $\chi_{1988}(307,·)$, $\chi_{1988}(1653,·)$, $\chi_{1988}(1457,·)$, $\chi_{1988}(953,·)$, $\chi_{1988}(1175,·)$, $\chi_{1988}(1987,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{245} a^{4} - \frac{1}{5}$, $\frac{1}{245} a^{5} - \frac{1}{5} a$, $\frac{1}{1715} a^{6} - \frac{1}{35} a^{2}$, $\frac{1}{1715} a^{7} - \frac{1}{35} a^{3}$, $\frac{1}{60025} a^{8} - \frac{2}{1225} a^{4} + \frac{1}{25}$, $\frac{1}{60025} a^{9} - \frac{2}{1225} a^{5} + \frac{1}{25} a$, $\frac{1}{10504375} a^{10} - \frac{4}{1500625} a^{8} - \frac{57}{214375} a^{6} - \frac{22}{30625} a^{4} + \frac{306}{4375} a^{2} + \frac{276}{625}$, $\frac{1}{178574375} a^{11} + \frac{146}{25510625} a^{9} + \frac{4}{214375} a^{7} - \frac{41}{30625} a^{5} + \frac{3931}{74375} a^{3} + \frac{176}{10625} a$, $\frac{1}{353755836875} a^{12} - \frac{908}{50536548125} a^{10} + \frac{27}{424676875} a^{8} - \frac{7257}{60668125} a^{6} + \frac{170769}{147336875} a^{4} - \frac{1488973}{21048125} a^{2} + \frac{42788}{176875}$, $\frac{1}{353755836875} a^{13} - \frac{59}{50536548125} a^{11} + \frac{4138}{7219506875} a^{9} - \frac{3861}{60668125} a^{7} - \frac{180434}{147336875} a^{5} - \frac{1158429}{21048125} a^{3} + \frac{151309}{601375} a$

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

Class group and class number

$C_{2}\times C_{336516}$, which has order $673032$ (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

\(\Q(\sqrt{-497}) \), 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$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}$$[2]^{7}$
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$71$71.14.13.1$x^{14} - 71$$14$$1$$13$$C_{14}$$[\ ]_{14}$