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Properties

Label 16.0.86874873937...8125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 59^{12}$
Root discriminant $64.37$
Ramified primes $5, 59$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1133189, -1197593, 2021597, -1838647, 1116794, -758009, 450586, -191415, 86380, -35165, 10869, -3398, 932, -172, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 44*x^14 - 172*x^13 + 932*x^12 - 3398*x^11 + 10869*x^10 - 35165*x^9 + 86380*x^8 - 191415*x^7 + 450586*x^6 - 758009*x^5 + 1116794*x^4 - 1838647*x^3 + 2021597*x^2 - 1197593*x + 1133189)
 
gp: K = bnfinit(x^16 - 4*x^15 + 44*x^14 - 172*x^13 + 932*x^12 - 3398*x^11 + 10869*x^10 - 35165*x^9 + 86380*x^8 - 191415*x^7 + 450586*x^6 - 758009*x^5 + 1116794*x^4 - 1838647*x^3 + 2021597*x^2 - 1197593*x + 1133189, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 44 x^{14} - 172 x^{13} + 932 x^{12} - 3398 x^{11} + 10869 x^{10} - 35165 x^{9} + 86380 x^{8} - 191415 x^{7} + 450586 x^{6} - 758009 x^{5} + 1116794 x^{4} - 1838647 x^{3} + 2021597 x^{2} - 1197593 x + 1133189 \)

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:  \(86874873937477183441455078125=5^{11}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.37$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{9} + \frac{1}{5} a^{7} + \frac{1}{3} a^{6} - \frac{1}{5} a^{5} - \frac{1}{3} a^{4} - \frac{2}{15} a^{3} - \frac{1}{3} a^{2} + \frac{2}{15} a + \frac{2}{5}$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{9} - \frac{2}{15} a^{8} - \frac{1}{15} a^{7} + \frac{2}{15} a^{6} - \frac{4}{15} a^{5} - \frac{7}{15} a^{4} + \frac{4}{15} a^{3} + \frac{2}{15} a^{2} + \frac{7}{15} a - \frac{2}{15}$, $\frac{1}{45} a^{12} + \frac{2}{15} a^{9} + \frac{4}{45} a^{8} - \frac{1}{45} a^{7} + \frac{7}{15} a^{6} - \frac{14}{45} a^{5} + \frac{1}{5} a^{4} - \frac{16}{45} a^{3} + \frac{7}{45} a^{2} - \frac{1}{5} a + \frac{14}{45}$, $\frac{1}{225} a^{13} - \frac{1}{225} a^{12} - \frac{1}{75} a^{10} - \frac{7}{45} a^{9} + \frac{11}{45} a^{8} - \frac{19}{45} a^{7} + \frac{2}{45} a^{6} - \frac{14}{45} a^{5} + \frac{22}{45} a^{4} - \frac{19}{225} a^{3} + \frac{59}{225} a^{2} - \frac{2}{45} a - \frac{98}{225}$, $\frac{1}{225} a^{14} - \frac{1}{225} a^{12} - \frac{1}{75} a^{11} + \frac{7}{225} a^{10} + \frac{7}{45} a^{9} + \frac{7}{45} a^{8} + \frac{2}{9} a^{7} - \frac{4}{15} a^{6} - \frac{4}{45} a^{5} + \frac{91}{225} a^{4} + \frac{4}{9} a^{3} - \frac{26}{225} a^{2} - \frac{31}{75} a + \frac{22}{225}$, $\frac{1}{937663063869557977627534336956146625} a^{15} + \frac{248649844419857327567343410223013}{937663063869557977627534336956146625} a^{14} + \frac{262519179414742546518133326208481}{187532612773911595525506867391229325} a^{13} + \frac{9583698565461347119521241745229883}{937663063869557977627534336956146625} a^{12} + \frac{7393971312021521223356301930597283}{937663063869557977627534336956146625} a^{11} + \frac{3521701358075628124211873872386031}{312554354623185992542511445652048875} a^{10} + \frac{4655844157735410866136346718340398}{62510870924637198508502289130409775} a^{9} + \frac{2618766988051926185570497955795585}{7501304510956463821020274695649173} a^{8} + \frac{2750299438201156066625079082709957}{5682806447694290773500208102764525} a^{7} + \frac{29671499090672733885922012832669573}{62510870924637198508502289130409775} a^{6} + \frac{128304586627483512561387320596952101}{937663063869557977627534336956146625} a^{5} - \frac{10440408408508010275083848944150246}{34728261624798443615834605072449875} a^{4} + \frac{84850842697772002761364215088205494}{187532612773911595525506867391229325} a^{3} - \frac{10934501770459190819263262384084204}{312554354623185992542511445652048875} a^{2} - \frac{187822591037387224475136553966129792}{937663063869557977627534336956146625} a - \frac{414362418806458389327422966529883702}{937663063869557977627534336956146625}$

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:  \( 56389577.6723 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 8.0.37866753125.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
59Data not computed