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Properties

Label 16.8.60210204307...3637.2
Degree $16$
Signature $[8, 4]$
Discriminant $3^{10}\cdot 13^{9}\cdot 1327^{8}$
Root discriminant $306.36$
Ramified primes $3, 13, 1327$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group 16T1675

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1729288237, -2443995613, 770369830, 634416836, -512361357, 89342780, 10820944, -8559267, 2494561, -38121, -46642, 15255, -2257, -65, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 - 65*x^13 - 2257*x^12 + 15255*x^11 - 46642*x^10 - 38121*x^9 + 2494561*x^8 - 8559267*x^7 + 10820944*x^6 + 89342780*x^5 - 512361357*x^4 + 634416836*x^3 + 770369830*x^2 - 2443995613*x + 1729288237)
 
gp: K = bnfinit(x^16 - 6*x^15 + 11*x^14 - 65*x^13 - 2257*x^12 + 15255*x^11 - 46642*x^10 - 38121*x^9 + 2494561*x^8 - 8559267*x^7 + 10820944*x^6 + 89342780*x^5 - 512361357*x^4 + 634416836*x^3 + 770369830*x^2 - 2443995613*x + 1729288237, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 11 x^{14} - 65 x^{13} - 2257 x^{12} + 15255 x^{11} - 46642 x^{10} - 38121 x^{9} + 2494561 x^{8} - 8559267 x^{7} + 10820944 x^{6} + 89342780 x^{5} - 512361357 x^{4} + 634416836 x^{3} + 770369830 x^{2} - 2443995613 x + 1729288237 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6021020430747726555546975715999071923637=3^{10}\cdot 13^{9}\cdot 1327^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.36$
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:  $3, 13, 1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} - \frac{1}{18} a^{6} - \frac{1}{9} a^{5} + \frac{7}{18} a^{4} + \frac{1}{18} a^{3} - \frac{7}{18} a^{2} - \frac{2}{9} a - \frac{7}{18}$, $\frac{1}{1812398329989038265097623840325660922122884656378025197403755982} a^{15} + \frac{33092156866212307885445654976985880334079453045338297278965459}{1812398329989038265097623840325660922122884656378025197403755982} a^{14} - \frac{46020046338022972960738533932615542095037888543881745857194939}{1812398329989038265097623840325660922122884656378025197403755982} a^{13} - \frac{55868005579095306289450836115793402741518685460680803621370225}{604132776663012755032541280108553640707628218792675065801251994} a^{12} - \frac{81135369790989397782419727357404773742146184416728703693781525}{604132776663012755032541280108553640707628218792675065801251994} a^{11} - \frac{44407392268790949187531684320013381373834211230667652055695765}{604132776663012755032541280108553640707628218792675065801251994} a^{10} + \frac{44175037347779860116528943109816435075246690348405273080921861}{906199164994519132548811920162830461061442328189012598701877991} a^{9} - \frac{92380271032638166434032356784007910420448450643609338804898638}{906199164994519132548811920162830461061442328189012598701877991} a^{8} - \frac{154137115610655172276992131545772955253768469794029435216826975}{1812398329989038265097623840325660922122884656378025197403755982} a^{7} + \frac{398148014726961082594907978391317225578644669853955727312936290}{906199164994519132548811920162830461061442328189012598701877991} a^{6} + \frac{218078221230994202071768376295346251493692002029935092069185159}{1812398329989038265097623840325660922122884656378025197403755982} a^{5} + \frac{80504612706427807398774119304195275989307795300339742111813595}{1812398329989038265097623840325660922122884656378025197403755982} a^{4} - \frac{355842014919147751482325233580017840087079741656890738216497765}{1812398329989038265097623840325660922122884656378025197403755982} a^{3} - \frac{449366917347015402246057309080737443439696521486029798592482426}{906199164994519132548811920162830461061442328189012598701877991} a^{2} + \frac{499777930353478505614837036769194593283232787447161152355084255}{1812398329989038265097623840325660922122884656378025197403755982} a - \frac{46890293297464336162322419138758592803006114387031990398723517}{302066388331506377516270640054276820353814109396337532900625997}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  $11$
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:  \( 1476738571660 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1327Data not computed