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Properties

Label 16.0.41937406987...921.21
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 89^{14}$
Root discriminant $168.43$
Ramified primes $11, 89$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21028770169, 0, -1883834854, 0, 57802141, 0, -707030, 0, -240908, 0, -5630, 0, 1917, 0, 50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169)
 
gp: K = bnfinit(x^16 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169, 1)
 

Normalized defining polynomial

\( x^{16} + 50 x^{14} + 1917 x^{12} - 5630 x^{10} - 240908 x^{8} - 707030 x^{6} + 57802141 x^{4} - 1883834854 x^{2} + 21028770169 \)

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:  \(419374069872350970710519002168327921=11^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.43$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{12} + \frac{1}{16} a^{6} - \frac{3}{32}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{29}{64} a - \frac{29}{64}$, $\frac{1}{191434477426460493552923070464} a^{14} - \frac{391570609478305107490167885}{191434477426460493552923070464} a^{12} - \frac{651329704811671024987321733}{11964654839153780847057691904} a^{10} - \frac{2982866950460413338206932519}{95717238713230246776461535232} a^{8} - \frac{1}{8} a^{7} + \frac{4428955366885330243931403475}{95717238713230246776461535232} a^{6} + \frac{644975263994705905194172205}{11964654839153780847057691904} a^{4} - \frac{9964645987097190860574589331}{191434477426460493552923070464} a^{2} + \frac{1}{8} a - \frac{8438269201763470017906902155}{17403134311496408504811188224}$, $\frac{1}{55520975750086631103180066434392064} a^{15} - \frac{1}{382868954852920987105846140928} a^{14} + \frac{321597565851004997082956210519731}{55520975750086631103180066434392064} a^{13} + \frac{391570609478305107490167885}{382868954852920987105846140928} a^{12} - \frac{187485297077389663321813137245925}{3470060984380414443948754152149504} a^{11} - \frac{844252150082551580894889755}{23929309678307561694115383808} a^{10} + \frac{1368035651441892841639238285370841}{27760487875043315551590033217196032} a^{9} + \frac{2982866950460413338206932519}{191434477426460493552923070464} a^{8} + \frac{366661276501234499388326899801555}{27760487875043315551590033217196032} a^{7} + \frac{19500354311422231450183980333}{191434477426460493552923070464} a^{6} + \frac{779161401828781616805390324133005}{3470060984380414443948754152149504} a^{5} + \frac{850606590899516700688039283}{23929309678307561694115383808} a^{4} - \frac{5969798425951444358391990065473939}{55520975750086631103180066434392064} a^{3} + \frac{9964645987097190860574589331}{382868954852920987105846140928} a^{2} - \frac{2089090470284280857117336897604491}{5047361431826057373016369675853824} a + \frac{4087485623889367891704105099}{34806268622992817009622376448}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, 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:  \( 59363790305.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{89}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-979}) \), 4.0.85301249.2, 4.4.704969.1, \(\Q(\sqrt{-11}, \sqrt{89})\), 8.0.647590974205440089.1 x2, 8.4.5351991522359009.3 x2, 8.0.7276303080960001.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$