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Properties

Label 14.0.61507729017...9168.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 3^{7}\cdot 7^{25}$
Root discriminant $158.20$
Ramified primes $2, 3, 7$
Class number $95596$ (GRH)
Class group $[2, 47798]$ (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![1959552, 0, 9144576, 0, 11430720, 0, 4889808, 0, 518616, 0, 19404, 0, 252, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 252*x^12 + 19404*x^10 + 518616*x^8 + 4889808*x^6 + 11430720*x^4 + 9144576*x^2 + 1959552)
 
gp: K = bnfinit(x^14 + 252*x^12 + 19404*x^10 + 518616*x^8 + 4889808*x^6 + 11430720*x^4 + 9144576*x^2 + 1959552, 1)
 

Normalized defining polynomial

\( x^{14} + 252 x^{12} + 19404 x^{10} + 518616 x^{8} + 4889808 x^{6} + 11430720 x^{4} + 9144576 x^{2} + 1959552 \)

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:  \(-6150772901711899500090300039168=-\,2^{21}\cdot 3^{7}\cdot 7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $158.20$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1176=2^{3}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1176}(1,·)$, $\chi_{1176}(83,·)$, $\chi_{1176}(1091,·)$, $\chi_{1176}(673,·)$, $\chi_{1176}(841,·)$, $\chi_{1176}(587,·)$, $\chi_{1176}(923,·)$, $\chi_{1176}(337,·)$, $\chi_{1176}(505,·)$, $\chi_{1176}(1009,·)$, $\chi_{1176}(419,·)$, $\chi_{1176}(755,·)$, $\chi_{1176}(169,·)$, $\chi_{1176}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{11671776} a^{10} - \frac{151}{972648} a^{8} - \frac{59}{81054} a^{6} - \frac{101}{18012} a^{4} - \frac{55}{9006} a^{2} + \frac{638}{1501}$, $\frac{1}{11671776} a^{11} - \frac{151}{972648} a^{9} - \frac{59}{81054} a^{7} - \frac{101}{18012} a^{5} - \frac{55}{9006} a^{3} + \frac{638}{1501} a$, $\frac{1}{2170950336} a^{12} + \frac{1}{40202784} a^{10} - \frac{683}{10050696} a^{8} + \frac{2927}{5025348} a^{6} - \frac{303}{186124} a^{4} + \frac{3517}{93062} a^{2} - \frac{16225}{46531}$, $\frac{1}{2170950336} a^{13} + \frac{1}{40202784} a^{11} - \frac{683}{10050696} a^{9} + \frac{2927}{5025348} a^{7} - \frac{303}{186124} a^{5} + \frac{3517}{93062} a^{3} - \frac{16225}{46531} a$

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

Class group and class number

$C_{2}\times C_{47798}$, which has order $95596$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-42}) \), 7.7.13841287201.1

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 R ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.40$x^{14} + 8 x^{13} - 4 x^{12} + 4 x^{11} + 5 x^{10} - 4 x^{9} - 4 x^{8} + 2 x^{7} - x^{6} + 6 x^{5} - 4 x^{4} + 6 x^{3} + 3 x^{2} + 6 x + 3$$2$$7$$21$$C_{14}$$[3]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.25.75$x^{14} + 112 x^{13} + 28 x^{12} - 98 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 161 x^{7} + 49 x^{6} + 147 x^{5} + 147 x^{2} - 147 x - 126$$14$$1$$25$$C_{14}$$[2]_{2}$