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Properties

Label 16.0.75677711862...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{6}$
Root discriminant $85.10$
Ramified primes $2, 5, 29, 89$
Class number $32$ (GRH)
Class group $[2, 16]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![276981329, 155779206, 8936718, 3593666, 10347492, 2452190, 90453, 180214, 46695, -8772, 1735, 312, -232, 32, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 + 32*x^13 - 232*x^12 + 312*x^11 + 1735*x^10 - 8772*x^9 + 46695*x^8 + 180214*x^7 + 90453*x^6 + 2452190*x^5 + 10347492*x^4 + 3593666*x^3 + 8936718*x^2 + 155779206*x + 276981329)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 + 32*x^13 - 232*x^12 + 312*x^11 + 1735*x^10 - 8772*x^9 + 46695*x^8 + 180214*x^7 + 90453*x^6 + 2452190*x^5 + 10347492*x^4 + 3593666*x^3 + 8936718*x^2 + 155779206*x + 276981329, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} + 32 x^{13} - 232 x^{12} + 312 x^{11} + 1735 x^{10} - 8772 x^{9} + 46695 x^{8} + 180214 x^{7} + 90453 x^{6} + 2452190 x^{5} + 10347492 x^{4} + 3593666 x^{3} + 8936718 x^{2} + 155779206 x + 276981329 \)

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:  \(7567771186285806117913600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.10$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} + \frac{5}{19} a^{13} - \frac{6}{19} a^{12} - \frac{5}{19} a^{11} - \frac{1}{19} a^{10} + \frac{1}{19} a^{9} + \frac{4}{19} a^{8} - \frac{4}{19} a^{7} + \frac{1}{19} a^{6} + \frac{2}{19} a^{5} + \frac{4}{19} a^{4} - \frac{6}{19} a^{3} + \frac{6}{19} a^{2} - \frac{6}{19} a + \frac{4}{19}$, $\frac{1}{20789350448212773338114118226846308232507242292186800899} a^{15} - \frac{408063727815069415895942281271670367367078538484669924}{20789350448212773338114118226846308232507242292186800899} a^{14} + \frac{5687672658815643374952234866400015372595978227909838433}{20789350448212773338114118226846308232507242292186800899} a^{13} + \frac{3918965964469229731825080627111149181760878165709219240}{20789350448212773338114118226846308232507242292186800899} a^{12} - \frac{6367313478607516467432842912840174131495578361100650549}{20789350448212773338114118226846308232507242292186800899} a^{11} + \frac{9769269502346253918327418385801565915499007370965327136}{20789350448212773338114118226846308232507242292186800899} a^{10} - \frac{7978524250602588102092205906140524760850001772059300545}{20789350448212773338114118226846308232507242292186800899} a^{9} + \frac{9005113892037487308703695132877875453913522642073663111}{20789350448212773338114118226846308232507242292186800899} a^{8} + \frac{4359220781594064003539045450262782287650700172363815485}{20789350448212773338114118226846308232507242292186800899} a^{7} - \frac{7489313723512496203449369867577984267070685426871598698}{20789350448212773338114118226846308232507242292186800899} a^{6} - \frac{665317748595461963854129823912399358109262672431610013}{20789350448212773338114118226846308232507242292186800899} a^{5} - \frac{733003859423985431127131955156875293819821100197163752}{1889940949837524848919465293349664384773385662926072809} a^{4} + \frac{4536335545474841613872189984675619408222116160033075545}{20789350448212773338114118226846308232507242292186800899} a^{3} + \frac{5129423480775553502093710737155888229527776953268398743}{20789350448212773338114118226846308232507242292186800899} a^{2} - \frac{7570606254497530018410677954323604077064564652937589979}{20789350448212773338114118226846308232507242292186800899} a - \frac{506545886000583690711810122712480429650048013565031737}{20789350448212773338114118226846308232507242292186800899}$

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

Class group and class number

$C_{2}\times C_{16}$, which has order $32$ (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:  \( 21064583.4527 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.0.11600.1, 4.0.35600.3, 8.0.1065849760000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$