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Properties

Label 16.0.14633639270...1237.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{9}\cdot 53^{14}$
Root discriminant $136.56$
Ramified primes $13, 53$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![600584543, -723490612, 740541568, -454458628, 248541840, -87176508, 21459942, -3540860, 137931, -65359, 52171, -12712, 1049, 98, -2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 2*x^14 + 98*x^13 + 1049*x^12 - 12712*x^11 + 52171*x^10 - 65359*x^9 + 137931*x^8 - 3540860*x^7 + 21459942*x^6 - 87176508*x^5 + 248541840*x^4 - 454458628*x^3 + 740541568*x^2 - 723490612*x + 600584543)
 
gp: K = bnfinit(x^16 - 5*x^15 - 2*x^14 + 98*x^13 + 1049*x^12 - 12712*x^11 + 52171*x^10 - 65359*x^9 + 137931*x^8 - 3540860*x^7 + 21459942*x^6 - 87176508*x^5 + 248541840*x^4 - 454458628*x^3 + 740541568*x^2 - 723490612*x + 600584543, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 2 x^{14} + 98 x^{13} + 1049 x^{12} - 12712 x^{11} + 52171 x^{10} - 65359 x^{9} + 137931 x^{8} - 3540860 x^{7} + 21459942 x^{6} - 87176508 x^{5} + 248541840 x^{4} - 454458628 x^{3} + 740541568 x^{2} - 723490612 x + 600584543 \)

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:  \(14633639270758674782026228182121237=13^{9}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.56$
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:  $13, 53$
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}{4023488696767245904079625686556934612952420861749172487695008120849} a^{15} + \frac{1716049002540492642489915124401364140065552239632745280817255827039}{4023488696767245904079625686556934612952420861749172487695008120849} a^{14} + \frac{722241970919468027240380586266045924469246755730025214791276396787}{4023488696767245904079625686556934612952420861749172487695008120849} a^{13} - \frac{1792973628768874878991702565295803299004300600963446429015246637868}{4023488696767245904079625686556934612952420861749172487695008120849} a^{12} - \frac{1539724277645502145408340434508713446995976775813413920938501931737}{4023488696767245904079625686556934612952420861749172487695008120849} a^{11} + \frac{1611318078569765176012777932906674488357930261027040599054084116339}{4023488696767245904079625686556934612952420861749172487695008120849} a^{10} - \frac{211300626682844571086990199912485997337721883577263729385507466074}{4023488696767245904079625686556934612952420861749172487695008120849} a^{9} - \frac{1083715119742858734669839725030676461602598979573916727441702441268}{4023488696767245904079625686556934612952420861749172487695008120849} a^{8} + \frac{1077993417242549630570047353163134442574266119114276065732153281167}{4023488696767245904079625686556934612952420861749172487695008120849} a^{7} + \frac{1253544059481068484858395669483807213919787454560598951223577502641}{4023488696767245904079625686556934612952420861749172487695008120849} a^{6} + \frac{591001526720302504059798479236163437580672106449494053478876202616}{4023488696767245904079625686556934612952420861749172487695008120849} a^{5} - \frac{1376859997518242873009770659011450113962148872479044466492117997119}{4023488696767245904079625686556934612952420861749172487695008120849} a^{4} - \frac{1131555633914204851402493914964097061239859147146367518860715621258}{4023488696767245904079625686556934612952420861749172487695008120849} a^{3} + \frac{394740645410139550977742086961861010343912903153762831896278978157}{4023488696767245904079625686556934612952420861749172487695008120849} a^{2} - \frac{52287674762099916414308795032866557096349977546528130615930007296}{4023488696767245904079625686556934612952420861749172487695008120849} a + \frac{1202845484245600106094521744170901299180969451598174831976521799989}{4023488696767245904079625686556934612952420861749172487695008120849}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( 686502773.431 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.3745777030801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$