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Properties

Label 14.0.51148327420...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 5^{7}\cdot 43^{12}$
Root discriminant $112.36$
Ramified primes $2, 5, 43$
Class number $28826$ (GRH)
Class group $[28826]$ (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![5334581, -3553304, 2734542, -1134722, 622438, -198708, 74769, -16892, 6239, -1044, 165, 46, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 46*x^11 + 165*x^10 - 1044*x^9 + 6239*x^8 - 16892*x^7 + 74769*x^6 - 198708*x^5 + 622438*x^4 - 1134722*x^3 + 2734542*x^2 - 3553304*x + 5334581)
 
gp: K = bnfinit(x^14 - 2*x^13 + 46*x^11 + 165*x^10 - 1044*x^9 + 6239*x^8 - 16892*x^7 + 74769*x^6 - 198708*x^5 + 622438*x^4 - 1134722*x^3 + 2734542*x^2 - 3553304*x + 5334581, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 46 x^{11} + 165 x^{10} - 1044 x^{9} + 6239 x^{8} - 16892 x^{7} + 74769 x^{6} - 198708 x^{5} + 622438 x^{4} - 1134722 x^{3} + 2734542 x^{2} - 3553304 x + 5334581 \)

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:  \(-51148327420496097793280000000=-\,2^{14}\cdot 5^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.36$
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:  $2, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(860=2^{2}\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{860}(1,·)$, $\chi_{860}(259,·)$, $\chi_{860}(379,·)$, $\chi_{860}(41,·)$, $\chi_{860}(821,·)$, $\chi_{860}(299,·)$, $\chi_{860}(59,·)$, $\chi_{860}(121,·)$, $\chi_{860}(661,·)$, $\chi_{860}(219,·)$, $\chi_{860}(279,·)$, $\chi_{860}(441,·)$, $\chi_{860}(699,·)$, $\chi_{860}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} + \frac{3}{49} a^{9} - \frac{3}{49} a^{8} + \frac{2}{49} a^{7} + \frac{1}{49} a^{6} - \frac{15}{49} a^{5} - \frac{1}{7} a^{4} - \frac{4}{49} a^{3} + \frac{18}{49} a^{2} + \frac{4}{49} a$, $\frac{1}{49} a^{11} + \frac{2}{49} a^{9} - \frac{3}{49} a^{8} + \frac{2}{49} a^{7} + \frac{3}{49} a^{6} + \frac{17}{49} a^{5} - \frac{11}{49} a^{4} - \frac{5}{49} a^{3} - \frac{15}{49} a^{2} + \frac{9}{49} a$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{9} + \frac{1}{49} a^{8} - \frac{1}{49} a^{7} + \frac{1}{49} a^{6} - \frac{16}{49} a^{5} - \frac{5}{49} a^{4} + \frac{15}{49} a^{2} + \frac{6}{49} a$, $\frac{1}{81567810384982339209450421} a^{13} + \frac{93501824685520881801865}{81567810384982339209450421} a^{12} + \frac{485293709601152750181054}{81567810384982339209450421} a^{11} + \frac{285612037468819396659015}{81567810384982339209450421} a^{10} + \frac{4449934284927841554512536}{81567810384982339209450421} a^{9} - \frac{408672279317082284605267}{81567810384982339209450421} a^{8} + \frac{1867553325642150534251347}{81567810384982339209450421} a^{7} - \frac{13348997248882375180035}{81567810384982339209450421} a^{6} + \frac{11176227622766563783610332}{81567810384982339209450421} a^{5} - \frac{35131871766614003847897252}{81567810384982339209450421} a^{4} - \frac{23052810882847578160009616}{81567810384982339209450421} a^{3} + \frac{36849764787236703828604716}{81567810384982339209450421} a^{2} + \frac{3750865231955796593712054}{81567810384982339209450421} a - \frac{168660016181271957958662}{1664649191530251820601029}$

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

Class group and class number

$C_{28826}$, which has order $28826$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-5}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$