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Properties

Label 16.0.52554919692...6256.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 17^{6}\cdot 89^{2}$
Root discriminant $26.31$
Ramified primes $2, 17, 89$
Class number $2$
Class group $[2]$
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1289, 1356, -1044, -688, 4323, 1652, -8840, -1476, 6994, -176, -2112, 84, 339, -8, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 28*x^14 - 8*x^13 + 339*x^12 + 84*x^11 - 2112*x^10 - 176*x^9 + 6994*x^8 - 1476*x^7 - 8840*x^6 + 1652*x^5 + 4323*x^4 - 688*x^3 - 1044*x^2 + 1356*x + 1289)
 
gp: K = bnfinit(x^16 - 28*x^14 - 8*x^13 + 339*x^12 + 84*x^11 - 2112*x^10 - 176*x^9 + 6994*x^8 - 1476*x^7 - 8840*x^6 + 1652*x^5 + 4323*x^4 - 688*x^3 - 1044*x^2 + 1356*x + 1289, 1)
 

Normalized defining polynomial

\( x^{16} - 28 x^{14} - 8 x^{13} + 339 x^{12} + 84 x^{11} - 2112 x^{10} - 176 x^{9} + 6994 x^{8} - 1476 x^{7} - 8840 x^{6} + 1652 x^{5} + 4323 x^{4} - 688 x^{3} - 1044 x^{2} + 1356 x + 1289 \)

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:  \(52554919692301559136256=2^{38}\cdot 17^{6}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{3}{13} a^{13} + \frac{3}{13} a^{12} - \frac{5}{13} a^{11} + \frac{4}{13} a^{10} + \frac{1}{13} a^{9} + \frac{1}{13} a^{8} - \frac{1}{13} a^{7} - \frac{1}{13} a^{6} + \frac{4}{13} a^{4} + \frac{2}{13} a^{3} - \frac{2}{13} a^{2} - \frac{1}{13} a - \frac{6}{13}$, $\frac{1}{410920388639455152361631} a^{15} - \frac{1445204303737506847793}{410920388639455152361631} a^{14} - \frac{115053412414658777410805}{410920388639455152361631} a^{13} + \frac{7944096267257636108409}{58702912662779307480233} a^{12} + \frac{76160317722324677648019}{410920388639455152361631} a^{11} - \frac{99938309949487099629375}{410920388639455152361631} a^{10} + \frac{159628320067380482811057}{410920388639455152361631} a^{9} - \frac{130217498395749763699606}{410920388639455152361631} a^{8} - \frac{71618854375773777042483}{410920388639455152361631} a^{7} + \frac{6023507043245959854894}{410920388639455152361631} a^{6} - \frac{26958849979644692408798}{58702912662779307480233} a^{5} - \frac{26662022620101338848841}{58702912662779307480233} a^{4} + \frac{107137000450103154170336}{410920388639455152361631} a^{3} + \frac{54170200010870555410619}{410920388639455152361631} a^{2} - \frac{148720185755106933232068}{410920388639455152361631} a + \frac{16504492019737168462245}{410920388639455152361631}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{254794939114071943288}{31609260664573473258587} a^{15} + \frac{86846158562482699182}{31609260664573473258587} a^{14} + \frac{6829110647244472465779}{31609260664573473258587} a^{13} - \frac{66317315679047634628}{4515608666367639036941} a^{12} - \frac{78987486452876435649851}{31609260664573473258587} a^{11} + \frac{12104202544838723860619}{31609260664573473258587} a^{10} + \frac{454679524649756306113000}{31609260664573473258587} a^{9} - \frac{175345027388461410099089}{31609260664573473258587} a^{8} - \frac{1291568559275436101623127}{31609260664573473258587} a^{7} + \frac{1055071588120230836835178}{31609260664573473258587} a^{6} + \frac{103023290829709567390250}{4515608666367639036941} a^{5} - \frac{99753147606603872114338}{4515608666367639036941} a^{4} - \frac{111444111771995695467832}{31609260664573473258587} a^{3} + \frac{214723149567430252249238}{31609260664573473258587} a^{2} + \frac{134214630584219154418531}{31609260664573473258587} a - \frac{232093567947851543021625}{31609260664573473258587} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 184965.134671 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.0.1088.2, 4.4.4352.1, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.22.87$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$