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Properties

Label 16.0.86874873937...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 59^{12}$
Root discriminant $64.37$
Ramified primes $5, 59$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![789479, -751719, -102285, 378341, 13538, -99480, -48, 19766, 7890, -2716, -953, 325, 168, 4, -20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 20*x^14 + 4*x^13 + 168*x^12 + 325*x^11 - 953*x^10 - 2716*x^9 + 7890*x^8 + 19766*x^7 - 48*x^6 - 99480*x^5 + 13538*x^4 + 378341*x^3 - 102285*x^2 - 751719*x + 789479)
 
gp: K = bnfinit(x^16 - 2*x^15 - 20*x^14 + 4*x^13 + 168*x^12 + 325*x^11 - 953*x^10 - 2716*x^9 + 7890*x^8 + 19766*x^7 - 48*x^6 - 99480*x^5 + 13538*x^4 + 378341*x^3 - 102285*x^2 - 751719*x + 789479, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 20 x^{14} + 4 x^{13} + 168 x^{12} + 325 x^{11} - 953 x^{10} - 2716 x^{9} + 7890 x^{8} + 19766 x^{7} - 48 x^{6} - 99480 x^{5} + 13538 x^{4} + 378341 x^{3} - 102285 x^{2} - 751719 x + 789479 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(86874873937477183441455078125=5^{11}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.37$
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:  $5, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{4918623566974145814466001276222092974064329} a^{15} - \frac{730375362285818419029149496378323869251119}{4918623566974145814466001276222092974064329} a^{14} - \frac{773438128707524417155127583912715257735458}{1639541188991381938155333758740697658021443} a^{13} + \frac{721443137858695011603075772117411911056158}{1639541188991381938155333758740697658021443} a^{12} - \frac{763849680524885440994286361450768015436116}{1639541188991381938155333758740697658021443} a^{11} + \frac{2021197087803849727683544207978109367982894}{4918623566974145814466001276222092974064329} a^{10} + \frac{1252451220490019652368163104351755100606776}{4918623566974145814466001276222092974064329} a^{9} + \frac{127375881797264298297998255002885739110444}{4918623566974145814466001276222092974064329} a^{8} - \frac{58777071910236338125291768844599436941681}{4918623566974145814466001276222092974064329} a^{7} - \frac{1159625210517541160895446913926469007420256}{4918623566974145814466001276222092974064329} a^{6} + \frac{1151628306010696878856245663530193485330281}{4918623566974145814466001276222092974064329} a^{5} + \frac{1239154652271789035449192358574838475524763}{4918623566974145814466001276222092974064329} a^{4} - \frac{1571308030938226725838959084042457568969144}{4918623566974145814466001276222092974064329} a^{3} + \frac{612051973102148362502704937621754121901181}{1639541188991381938155333758740697658021443} a^{2} + \frac{166324520852924092505017956607174381593068}{702660509567735116352285896603156139152047} a + \frac{366539488133363550726441658928021576559748}{1639541188991381938155333758740697658021443}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $7$
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:  \( 25352074.1318 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 8.0.37866753125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
59Data not computed