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Properties

Label 16.4.33402063455...1009.1
Degree $16$
Signature $[4, 6]$
Discriminant $23^{6}\cdot 41^{12}$
Root discriminant $52.51$
Ramified primes $23, 41$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-150064, 467792, -230872, -276266, 204683, 94011, -114234, 18828, 13605, -6988, 167, 895, -307, 9, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 + 9*x^13 - 307*x^12 + 895*x^11 + 167*x^10 - 6988*x^9 + 13605*x^8 + 18828*x^7 - 114234*x^6 + 94011*x^5 + 204683*x^4 - 276266*x^3 - 230872*x^2 + 467792*x - 150064)
 
gp: K = bnfinit(x^16 - 6*x^15 + 21*x^14 + 9*x^13 - 307*x^12 + 895*x^11 + 167*x^10 - 6988*x^9 + 13605*x^8 + 18828*x^7 - 114234*x^6 + 94011*x^5 + 204683*x^4 - 276266*x^3 - 230872*x^2 + 467792*x - 150064, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 21 x^{14} + 9 x^{13} - 307 x^{12} + 895 x^{11} + 167 x^{10} - 6988 x^{9} + 13605 x^{8} + 18828 x^{7} - 114234 x^{6} + 94011 x^{5} + 204683 x^{4} - 276266 x^{3} - 230872 x^{2} + 467792 x - 150064 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{46} a^{10} + \frac{9}{46} a^{9} - \frac{5}{23} a^{8} - \frac{13}{46} a^{7} + \frac{21}{46} a^{6} - \frac{7}{46} a^{5} - \frac{7}{23} a^{4} - \frac{11}{46} a^{3} + \frac{19}{46} a^{2} - \frac{7}{23} a - \frac{2}{23}$, $\frac{1}{230} a^{11} + \frac{1}{230} a^{10} + \frac{1}{23} a^{9} - \frac{5}{46} a^{8} - \frac{59}{230} a^{7} + \frac{11}{46} a^{6} + \frac{44}{115} a^{5} - \frac{83}{230} a^{4} - \frac{31}{230} a^{3} - \frac{37}{115} a^{2} + \frac{8}{115} a - \frac{7}{115}$, $\frac{1}{230} a^{12} - \frac{1}{230} a^{10} - \frac{1}{23} a^{9} - \frac{49}{230} a^{8} + \frac{7}{115} a^{7} - \frac{31}{115} a^{6} + \frac{7}{115} a^{5} + \frac{77}{230} a^{4} + \frac{67}{230} a^{3} + \frac{3}{46} a^{2} - \frac{1}{46} a + \frac{27}{115}$, $\frac{1}{230} a^{13} + \frac{1}{230} a^{10} + \frac{51}{230} a^{9} - \frac{111}{230} a^{8} - \frac{21}{230} a^{7} + \frac{49}{230} a^{6} + \frac{19}{46} a^{5} + \frac{37}{115} a^{4} + \frac{52}{115} a^{3} + \frac{111}{230} a^{2} - \frac{7}{23} a - \frac{27}{115}$, $\frac{1}{52900} a^{14} - \frac{24}{13225} a^{13} + \frac{87}{52900} a^{12} - \frac{67}{52900} a^{11} + \frac{31}{10580} a^{10} + \frac{3423}{52900} a^{9} + \frac{7627}{52900} a^{8} + \frac{1284}{2645} a^{7} + \frac{11077}{52900} a^{6} - \frac{4218}{13225} a^{5} - \frac{6548}{13225} a^{4} - \frac{19641}{52900} a^{3} - \frac{3739}{52900} a^{2} - \frac{2688}{13225} a - \frac{1384}{13225}$, $\frac{1}{7944336744982829736187563400} a^{15} - \frac{35312809079548054736391}{3972168372491414868093781700} a^{14} - \frac{1746448961228259446299527}{7944336744982829736187563400} a^{13} + \frac{11312830307019645529002461}{7944336744982829736187563400} a^{12} + \frac{2247984389248773524115577}{7944336744982829736187563400} a^{11} - \frac{8577896007196478261389177}{7944336744982829736187563400} a^{10} + \frac{1646631086587629340331778079}{7944336744982829736187563400} a^{9} + \frac{439491400567231326655058236}{993042093122853717023445425} a^{8} - \frac{18969237593461366269492783}{7944336744982829736187563400} a^{7} + \frac{416558127094792085911915232}{993042093122853717023445425} a^{6} - \frac{54108148356950530161617073}{794433674498282973618756340} a^{5} - \frac{1236790849954562854735224269}{7944336744982829736187563400} a^{4} - \frac{3586474657648947503180530633}{7944336744982829736187563400} a^{3} + \frac{418987453970802886458934221}{3972168372491414868093781700} a^{2} - \frac{735290908182141853760584301}{1986084186245707434046890850} a + \frac{889979833289014479563344}{8787983124981006345340225}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 5487683.59986 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.2.38663.1, 4.2.1585183.1, 8.2.57794518300247.1, 8.2.34381034087.1, 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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ 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 ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$