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Properties

Label 16.4.34659014039...801.17
Degree $16$
Signature $[4, 6]$
Discriminant $11^{6}\cdot 89^{14}$
Root discriminant $124.81$
Ramified primes $11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167431328, 14876208, -20716338, -22671797, 12050174, -16201717, 4112201, -1374433, 129601, 59261, -15475, 8121, -861, 117, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 3*x^14 + 117*x^13 - 861*x^12 + 8121*x^11 - 15475*x^10 + 59261*x^9 + 129601*x^8 - 1374433*x^7 + 4112201*x^6 - 16201717*x^5 + 12050174*x^4 - 22671797*x^3 - 20716338*x^2 + 14876208*x + 167431328)
 
gp: K = bnfinit(x^16 - 4*x^15 - 3*x^14 + 117*x^13 - 861*x^12 + 8121*x^11 - 15475*x^10 + 59261*x^9 + 129601*x^8 - 1374433*x^7 + 4112201*x^6 - 16201717*x^5 + 12050174*x^4 - 22671797*x^3 - 20716338*x^2 + 14876208*x + 167431328, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 3 x^{14} + 117 x^{13} - 861 x^{12} + 8121 x^{11} - 15475 x^{10} + 59261 x^{9} + 129601 x^{8} - 1374433 x^{7} + 4112201 x^{6} - 16201717 x^{5} + 12050174 x^{4} - 22671797 x^{3} - 20716338 x^{2} + 14876208 x + 167431328 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3465901403903727030665446298911801=11^{6}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.81$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{5}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{5}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{176} a^{13} + \frac{5}{176} a^{12} - \frac{3}{176} a^{11} + \frac{1}{176} a^{10} + \frac{7}{176} a^{9} - \frac{13}{176} a^{8} + \frac{25}{176} a^{7} + \frac{39}{176} a^{6} + \frac{31}{176} a^{5} + \frac{27}{176} a^{4} + \frac{21}{176} a^{3} - \frac{15}{176} a^{2} - \frac{1}{8} a + \frac{9}{22}$, $\frac{1}{704} a^{14} + \frac{1}{704} a^{13} + \frac{21}{704} a^{12} + \frac{13}{704} a^{11} - \frac{41}{704} a^{10} - \frac{41}{704} a^{9} + \frac{3}{64} a^{8} - \frac{61}{704} a^{7} + \frac{95}{704} a^{6} - \frac{97}{704} a^{5} - \frac{131}{704} a^{4} - \frac{17}{64} a^{3} + \frac{129}{352} a^{2} + \frac{31}{88} a + \frac{1}{11}$, $\frac{1}{63211058514102373385277225599594454770120311688208448} a^{15} + \frac{22715327948154630684014079798407157809317649427179}{63211058514102373385277225599594454770120311688208448} a^{14} + \frac{79023823543270390144750204699289758217053998037663}{63211058514102373385277225599594454770120311688208448} a^{13} + \frac{457219995222169003944256093225936433159543865989319}{63211058514102373385277225599594454770120311688208448} a^{12} - \frac{1576652431957725193238639294059681526805882017311015}{63211058514102373385277225599594454770120311688208448} a^{11} + \frac{7638414605080380977338909748693900586037977295720301}{63211058514102373385277225599594454770120311688208448} a^{10} - \frac{1601503785714446086103557511552786061751989334495265}{63211058514102373385277225599594454770120311688208448} a^{9} + \frac{2957974232924957270143332034304271186591703974693357}{63211058514102373385277225599594454770120311688208448} a^{8} - \frac{3151040843806465912076930457589046705314782723054211}{63211058514102373385277225599594454770120311688208448} a^{7} - \frac{7976547843459600226825670928175603685943796943770291}{63211058514102373385277225599594454770120311688208448} a^{6} + \frac{941485718847544740199516598560120074327603677907251}{63211058514102373385277225599594454770120311688208448} a^{5} + \frac{1255064577006599817555623077501701903022225040459511}{63211058514102373385277225599594454770120311688208448} a^{4} + \frac{697133771022970304782651720663804694598685940490975}{15802764628525593346319306399898613692530077922052112} a^{3} + \frac{7456620841731155236874465859807397475662004136678515}{15802764628525593346319306399898613692530077922052112} a^{2} - \frac{779687330485166916185878445458638533325858286367557}{3950691157131398336579826599974653423132519480513028} a - \frac{256423069206139364878841530306528032562898575285523}{987672789282849584144956649993663355783129870128257}$

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:  $9$
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:  \( 1832140294650 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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

\(\Q(\sqrt{89}) \), 4.2.87131.1, 4.4.704969.1, 4.2.7754659.1, 8.4.5351991522359009.3 x2, 8.4.60134736206281.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ 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$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$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}$