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Properties

Label 16.0.35124794539...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{12}\cdot 37^{8}$
Root discriminant $34.21$
Ramified primes $2, 5, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![233456, 46280, -185116, -10468, 105348, -102864, 35844, 5804, -12036, 6826, -1482, -10, 292, -112, 39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 39*x^14 - 112*x^13 + 292*x^12 - 10*x^11 - 1482*x^10 + 6826*x^9 - 12036*x^8 + 5804*x^7 + 35844*x^6 - 102864*x^5 + 105348*x^4 - 10468*x^3 - 185116*x^2 + 46280*x + 233456)
 
gp: K = bnfinit(x^16 - 6*x^15 + 39*x^14 - 112*x^13 + 292*x^12 - 10*x^11 - 1482*x^10 + 6826*x^9 - 12036*x^8 + 5804*x^7 + 35844*x^6 - 102864*x^5 + 105348*x^4 - 10468*x^3 - 185116*x^2 + 46280*x + 233456, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 39 x^{14} - 112 x^{13} + 292 x^{12} - 10 x^{11} - 1482 x^{10} + 6826 x^{9} - 12036 x^{8} + 5804 x^{7} + 35844 x^{6} - 102864 x^{5} + 105348 x^{4} - 10468 x^{3} - 185116 x^{2} + 46280 x + 233456 \)

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:  \(3512479453921000000000000=2^{12}\cdot 5^{12}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.21$
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:  $2, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{1929813197575310594023635567351954232} a^{15} + \frac{22807963477060085694182218078363927}{241226649696913824252954445918994279} a^{14} + \frac{111186588063142039942505363671860139}{1929813197575310594023635567351954232} a^{13} + \frac{6839180012857170091563889282571943}{964906598787655297011817783675977116} a^{12} + \frac{31581610226775892896414700509180369}{482453299393827648505908891837988558} a^{11} - \frac{132014379353237945049231806806485125}{964906598787655297011817783675977116} a^{10} - \frac{215905762279526220810884112144898647}{964906598787655297011817783675977116} a^{9} - \frac{171974065016086238196918411998133421}{964906598787655297011817783675977116} a^{8} + \frac{6773512620958511147981124643397999}{241226649696913824252954445918994279} a^{7} - \frac{28186127658161765826255865294467549}{241226649696913824252954445918994279} a^{6} + \frac{169615581219093543493354452588274441}{482453299393827648505908891837988558} a^{5} + \frac{45568126451064364668333999357322509}{241226649696913824252954445918994279} a^{4} - \frac{11061540296588680647114167432837237}{482453299393827648505908891837988558} a^{3} - \frac{80721266090052353041949585890566599}{482453299393827648505908891837988558} a^{2} + \frac{24949967085182864920793548319987867}{482453299393827648505908891837988558} a + \frac{42977644485978188930313303187756524}{241226649696913824252954445918994279}$

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:  \( 137097.711884 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3700.1, 8.0.13690000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
5Data not computed
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$