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Properties

Label 16.4.33402063455...009.13
Degree $16$
Signature $[4, 6]$
Discriminant $23^{6}\cdot 41^{12}$
Root discriminant $52.51$
Ramified primes $23, 41$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11839, -10598, -12852, 51574, -36601, 9964, 3590, -10256, 9300, -6742, 3962, -1904, 742, -224, 52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 742*x^12 - 1904*x^11 + 3962*x^10 - 6742*x^9 + 9300*x^8 - 10256*x^7 + 3590*x^6 + 9964*x^5 - 36601*x^4 + 51574*x^3 - 12852*x^2 - 10598*x + 11839)
 
gp: K = bnfinit(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 742*x^12 - 1904*x^11 + 3962*x^10 - 6742*x^9 + 9300*x^8 - 10256*x^7 + 3590*x^6 + 9964*x^5 - 36601*x^4 + 51574*x^3 - 12852*x^2 - 10598*x + 11839, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 742 x^{12} - 1904 x^{11} + 3962 x^{10} - 6742 x^{9} + 9300 x^{8} - 10256 x^{7} + 3590 x^{6} + 9964 x^{5} - 36601 x^{4} + 51574 x^{3} - 12852 x^{2} - 10598 x + 11839 \)

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:  \(3340206345557585382040261009=23^{6}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.51$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a - \frac{1}{12}$, $\frac{1}{12} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{12} a^{3} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{7} - \frac{1}{4} a^{4} - \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{6} a^{7} + \frac{1}{12} a^{5} + \frac{1}{6} a^{4} + \frac{5}{12} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{5664} a^{12} - \frac{1}{944} a^{11} - \frac{17}{944} a^{10} + \frac{31}{1888} a^{9} - \frac{103}{5664} a^{8} - \frac{289}{1416} a^{7} + \frac{595}{5664} a^{6} - \frac{5}{1416} a^{5} + \frac{371}{1888} a^{4} + \frac{1117}{5664} a^{3} + \frac{925}{1888} a^{2} - \frac{17}{96} a - \frac{1045}{5664}$, $\frac{1}{5664} a^{13} - \frac{23}{944} a^{11} - \frac{47}{5664} a^{10} - \frac{17}{5664} a^{9} + \frac{19}{944} a^{8} + \frac{1211}{5664} a^{7} - \frac{113}{2832} a^{6} - \frac{895}{5664} a^{5} + \frac{715}{5664} a^{4} + \frac{1453}{5664} a^{3} - \frac{291}{1888} a^{2} + \frac{635}{1888} a + \frac{169}{2832}$, $\frac{1}{4810033056} a^{14} - \frac{7}{4810033056} a^{13} - \frac{46237}{4810033056} a^{12} + \frac{277513}{4810033056} a^{11} + \frac{2999303}{801672176} a^{10} - \frac{15420521}{801672176} a^{9} - \frac{11145427}{2405016528} a^{8} + \frac{647355101}{4810033056} a^{7} - \frac{454718341}{2405016528} a^{6} + \frac{738720}{50104511} a^{5} - \frac{465660715}{4810033056} a^{4} + \frac{584982191}{1603344352} a^{3} + \frac{199399715}{4810033056} a^{2} - \frac{300233555}{1202508264} a - \frac{217800877}{1603344352}$, $\frac{1}{4728262494048} a^{15} + \frac{121}{1182065623512} a^{14} - \frac{320209007}{4728262494048} a^{13} - \frac{28369457}{4728262494048} a^{12} + \frac{36756024601}{4728262494048} a^{11} - \frac{130726304387}{4728262494048} a^{10} + \frac{29022196255}{2364131247024} a^{9} + \frac{10808045385}{788043749008} a^{8} + \frac{86619711807}{788043749008} a^{7} + \frac{344700392889}{1576087498016} a^{6} - \frac{125274669047}{1182065623512} a^{5} - \frac{167674845415}{2364131247024} a^{4} + \frac{256674211855}{788043749008} a^{3} + \frac{71534828791}{1576087498016} a^{2} + \frac{482953322355}{1576087498016} a + \frac{231891131483}{591032811756}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 14663707.1604 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.38663.1, 4.4.68921.1, 4.2.1585183.1, 8.4.2512805143489.1

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 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$