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Properties

Label 16.0.56009470234...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{2}\cdot 101^{4}$
Root discriminant $26.41$
Ramified primes $2, 5, 29, 101$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3280, 4240, 3640, 4560, 3376, 2092, 1788, 584, 513, 296, -42, 112, 7, -12, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 12*x^13 + 7*x^12 + 112*x^11 - 42*x^10 + 296*x^9 + 513*x^8 + 584*x^7 + 1788*x^6 + 2092*x^5 + 3376*x^4 + 4560*x^3 + 3640*x^2 + 4240*x + 3280)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 12*x^13 + 7*x^12 + 112*x^11 - 42*x^10 + 296*x^9 + 513*x^8 + 584*x^7 + 1788*x^6 + 2092*x^5 + 3376*x^4 + 4560*x^3 + 3640*x^2 + 4240*x + 3280, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 12 x^{13} + 7 x^{12} + 112 x^{11} - 42 x^{10} + 296 x^{9} + 513 x^{8} + 584 x^{7} + 1788 x^{6} + 2092 x^{5} + 3376 x^{4} + 4560 x^{3} + 3640 x^{2} + 4240 x + 3280 \)

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:  \(56009470234240000000000=2^{16}\cdot 5^{10}\cdot 29^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.41$
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:  $2, 5, 29, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5}$, $\frac{1}{24} a^{12} - \frac{1}{8} a^{11} - \frac{1}{12} a^{10} + \frac{1}{24} a^{9} + \frac{1}{8} a^{7} + \frac{5}{24} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{11} + \frac{1}{24} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{6} a^{7} + \frac{1}{8} a^{6} - \frac{1}{12} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{120} a^{14} - \frac{1}{60} a^{11} - \frac{1}{120} a^{10} - \frac{1}{30} a^{9} - \frac{1}{30} a^{8} - \frac{1}{5} a^{7} - \frac{7}{120} a^{6} - \frac{3}{20} a^{5} + \frac{7}{60} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{101491223945660520} a^{15} + \frac{13205659134787}{12686402993207565} a^{14} + \frac{29430260562825}{1691520399094342} a^{13} - \frac{119940984848017}{16915203990943420} a^{12} - \frac{3246440997650963}{101491223945660520} a^{11} + \frac{15739744400923}{247539570599172} a^{10} + \frac{5465716676669251}{50745611972830260} a^{9} - \frac{2545312820920007}{25372805986415130} a^{8} + \frac{4805425751028419}{101491223945660520} a^{7} - \frac{2417713259288803}{10149122394566052} a^{6} - \frac{17151709693558457}{50745611972830260} a^{5} + \frac{683921319086011}{2985035998401780} a^{4} + \frac{840133577805283}{1691520399094342} a^{3} - \frac{572253840206527}{5074561197283026} a^{2} - \frac{48574989163904}{149251799920089} a - \frac{9950792751495}{20628297549931}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( \frac{3240294223}{2905144523992} a^{15} - \frac{1089070095}{363143065499} a^{14} + \frac{12910559635}{1452572261996} a^{13} + \frac{4869908275}{726286130998} a^{12} - \frac{27724027795}{2905144523992} a^{11} + \frac{1983316763}{17714295878} a^{10} + \frac{129577261955}{1452572261996} a^{9} + \frac{193133410375}{726286130998} a^{8} + \frac{1650442394395}{2905144523992} a^{7} + \frac{377326230030}{363143065499} a^{6} + \frac{735232544886}{363143065499} a^{5} + \frac{854521761270}{363143065499} a^{4} + \frac{2583415821355}{726286130998} a^{3} + \frac{1402290879930}{363143065499} a^{2} + \frac{1060816970055}{363143065499} a + \frac{11820396477}{8857147939} \) (order $4$)
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:  \( 35770.9790657 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.4.2525.1, 4.0.40400.1, \(\Q(i, \sqrt{5})\), 8.0.1632160000.5

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$