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Properties

Label 16.0.95735899582...8944.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{12}$
Root discriminant $23.65$
Ramified primes $2, 3$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![433, -344, 1048, 448, -184, -440, 280, -160, 310, -72, -8, 0, 40, -8, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 8*x^13 + 40*x^12 - 8*x^10 - 72*x^9 + 310*x^8 - 160*x^7 + 280*x^6 - 440*x^5 - 184*x^4 + 448*x^3 + 1048*x^2 - 344*x + 433)
 
gp: K = bnfinit(x^16 - 8*x^14 - 8*x^13 + 40*x^12 - 8*x^10 - 72*x^9 + 310*x^8 - 160*x^7 + 280*x^6 - 440*x^5 - 184*x^4 + 448*x^3 + 1048*x^2 - 344*x + 433, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 8 x^{13} + 40 x^{12} - 8 x^{10} - 72 x^{9} + 310 x^{8} - 160 x^{7} + 280 x^{6} - 440 x^{5} - 184 x^{4} + 448 x^{3} + 1048 x^{2} - 344 x + 433 \)

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:  \(9573589958277615058944=2^{54}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{28} a^{12} - \frac{1}{14} a^{11} - \frac{3}{14} a^{10} + \frac{1}{28} a^{8} - \frac{1}{14} a^{7} + \frac{1}{7} a^{6} - \frac{1}{2} a^{5} - \frac{1}{28} a^{4} - \frac{1}{2} a^{2} + \frac{1}{14} a - \frac{9}{28}$, $\frac{1}{28} a^{13} - \frac{3}{28} a^{11} - \frac{5}{28} a^{10} - \frac{3}{14} a^{9} - \frac{1}{4} a^{8} - \frac{3}{14} a^{6} - \frac{1}{28} a^{5} - \frac{1}{14} a^{4} + \frac{1}{4} a^{3} - \frac{5}{28} a^{2} + \frac{1}{14} a - \frac{11}{28}$, $\frac{1}{24467548} a^{14} - \frac{836}{6116887} a^{13} + \frac{70513}{6116887} a^{12} + \frac{2307377}{24467548} a^{11} + \frac{388711}{12233774} a^{10} - \frac{700229}{24467548} a^{9} + \frac{2966807}{24467548} a^{8} + \frac{183810}{873841} a^{7} + \frac{5006053}{24467548} a^{6} - \frac{2060024}{6116887} a^{5} + \frac{538037}{12233774} a^{4} + \frac{5022243}{24467548} a^{3} + \frac{1392158}{6116887} a^{2} - \frac{3956399}{24467548} a + \frac{11985475}{24467548}$, $\frac{1}{1004516969271004} a^{15} - \frac{1023653}{502258484635502} a^{14} - \frac{16452137333743}{1004516969271004} a^{13} + \frac{882416483539}{1004516969271004} a^{12} + \frac{125505634360801}{1004516969271004} a^{11} + \frac{64335829489383}{502258484635502} a^{10} - \frac{4522763846541}{1004516969271004} a^{9} + \frac{20016019039215}{1004516969271004} a^{8} - \frac{15441176795423}{143502424181572} a^{7} - \frac{21356968442097}{502258484635502} a^{6} - \frac{27949284266155}{143502424181572} a^{5} + \frac{203389673246885}{1004516969271004} a^{4} + \frac{41400302091587}{1004516969271004} a^{3} - \frac{12556303659123}{71751212090786} a^{2} - \frac{50901112545151}{1004516969271004} a + \frac{396934206925341}{1004516969271004}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 10073.1288883 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.0.18432.2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.2048.2, 4.4.27648.1 x2, 4.4.13824.1 x2, 8.0.1358954496.3, 8.8.3057647616.1, 8.0.12230590464.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed