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Properties

Label 16.0.88295032325...5913.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{15}\cdot 29^{7}$
Root discriminant $48.32$
Ramified primes $13, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![481807, 1067861, 1152161, 878457, 431951, 176475, 111618, 41171, 8970, 5057, 1846, 286, 78, 44, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 10*x^14 + 44*x^13 + 78*x^12 + 286*x^11 + 1846*x^10 + 5057*x^9 + 8970*x^8 + 41171*x^7 + 111618*x^6 + 176475*x^5 + 431951*x^4 + 878457*x^3 + 1152161*x^2 + 1067861*x + 481807)
 
gp: K = bnfinit(x^16 - 2*x^15 + 10*x^14 + 44*x^13 + 78*x^12 + 286*x^11 + 1846*x^10 + 5057*x^9 + 8970*x^8 + 41171*x^7 + 111618*x^6 + 176475*x^5 + 431951*x^4 + 878457*x^3 + 1152161*x^2 + 1067861*x + 481807, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 10 x^{14} + 44 x^{13} + 78 x^{12} + 286 x^{11} + 1846 x^{10} + 5057 x^{9} + 8970 x^{8} + 41171 x^{7} + 111618 x^{6} + 176475 x^{5} + 431951 x^{4} + 878457 x^{3} + 1152161 x^{2} + 1067861 x + 481807 \)

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:  \(882950323258772752350175913=13^{15}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.32$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{85536160631659962068067434138579001549861} a^{15} + \frac{27758748801683938260981833395425867858131}{85536160631659962068067434138579001549861} a^{14} + \frac{7467100707861507165884889060949475609703}{28512053543886654022689144712859667183287} a^{13} - \frac{33349413828195022482219092041562919404581}{85536160631659962068067434138579001549861} a^{12} - \frac{39614868510565380128996757686951052298276}{85536160631659962068067434138579001549861} a^{11} + \frac{852108780301627614139322896544410828668}{28512053543886654022689144712859667183287} a^{10} - \frac{2523908716149090028421033800683651757475}{85536160631659962068067434138579001549861} a^{9} + \frac{7002710462396910620623462163661499173082}{28512053543886654022689144712859667183287} a^{8} - \frac{14206343784200643374994094384933130911033}{28512053543886654022689144712859667183287} a^{7} + \frac{27751258997990687019914885923161622931453}{85536160631659962068067434138579001549861} a^{6} + \frac{24092632920983566449366476543501118227156}{85536160631659962068067434138579001549861} a^{5} + \frac{11095093221853807135575853659367582399175}{85536160631659962068067434138579001549861} a^{4} - \frac{24516166101492645134348551921013983126277}{85536160631659962068067434138579001549861} a^{3} + \frac{41928049404557960436596756205338630836936}{85536160631659962068067434138579001549861} a^{2} + \frac{4739756240158899593686626356961720535157}{28512053543886654022689144712859667183287} a - \frac{17325716732844680703932121224683609177282}{85536160631659962068067434138579001549861}$

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:  \( 1109982.96591 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.1819706993.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $16$

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
13Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$