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Properties

Label 16.0.38871673299...7769.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 37^{12}$
Root discriminant $29.81$
Ramified primes $3, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2673, -1053, 162, -387, -939, 54, -422, 384, 133, 56, 85, -4, 10, -22, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 22*x^13 + 10*x^12 - 4*x^11 + 85*x^10 + 56*x^9 + 133*x^8 + 384*x^7 - 422*x^6 + 54*x^5 - 939*x^4 - 387*x^3 + 162*x^2 - 1053*x + 2673)
 
gp: K = bnfinit(x^16 - 2*x^14 - 22*x^13 + 10*x^12 - 4*x^11 + 85*x^10 + 56*x^9 + 133*x^8 + 384*x^7 - 422*x^6 + 54*x^5 - 939*x^4 - 387*x^3 + 162*x^2 - 1053*x + 2673, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} - 22 x^{13} + 10 x^{12} - 4 x^{11} + 85 x^{10} + 56 x^{9} + 133 x^{8} + 384 x^{7} - 422 x^{6} + 54 x^{5} - 939 x^{4} - 387 x^{3} + 162 x^{2} - 1053 x + 2673 \)

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:  \(388716732992848243307769=3^{10}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.81$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{27} a^{8} + \frac{4}{27} a^{7} + \frac{1}{9} a^{6} + \frac{1}{27} a^{5} - \frac{4}{27} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} - \frac{2}{27} a^{7} - \frac{2}{27} a^{6} + \frac{1}{9} a^{5} - \frac{10}{27} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{4}{27} a^{7} + \frac{1}{9} a^{5} - \frac{13}{27} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{1}{81} a^{12} + \frac{1}{81} a^{10} + \frac{1}{81} a^{9} + \frac{2}{81} a^{7} - \frac{1}{81} a^{6} + \frac{1}{81} a^{5} + \frac{1}{3} a^{4} + \frac{8}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{2644200722097063} a^{15} - \frac{5047080241826}{881400240699021} a^{14} - \frac{28201243828463}{2644200722097063} a^{13} + \frac{10646375159996}{2644200722097063} a^{12} - \frac{41156840540783}{2644200722097063} a^{11} - \frac{129892071897826}{2644200722097063} a^{10} - \frac{62313805695383}{2644200722097063} a^{9} + \frac{114165743821682}{2644200722097063} a^{8} + \frac{133526637984754}{2644200722097063} a^{7} + \frac{562464778238}{32644453359223} a^{6} + \frac{139918262194711}{2644200722097063} a^{5} + \frac{54578114928727}{293800080233007} a^{4} - \frac{49197321804370}{881400240699021} a^{3} + \frac{38466926368256}{293800080233007} a^{2} + \frac{10042614187610}{97933360077669} a + \frac{15608158174594}{32644453359223}$

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:  \( 342861.060416 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.2.4107.1, 8.0.5616860517.1, 8.2.69274613043.1, 8.2.16850581551.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
37Data not computed