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Properties

Label 16.0.74256122186...2833.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{15}\cdot 29^{9}$
Root discriminant $73.61$
Ramified primes $13, 29$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![836007, -2046301, 2213979, -1586595, 1106547, -680316, 296361, -130078, 57538, -16926, 5655, -1846, 364, -116, 27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 27*x^14 - 116*x^13 + 364*x^12 - 1846*x^11 + 5655*x^10 - 16926*x^9 + 57538*x^8 - 130078*x^7 + 296361*x^6 - 680316*x^5 + 1106547*x^4 - 1586595*x^3 + 2213979*x^2 - 2046301*x + 836007)
 
gp: K = bnfinit(x^16 - 4*x^15 + 27*x^14 - 116*x^13 + 364*x^12 - 1846*x^11 + 5655*x^10 - 16926*x^9 + 57538*x^8 - 130078*x^7 + 296361*x^6 - 680316*x^5 + 1106547*x^4 - 1586595*x^3 + 2213979*x^2 - 2046301*x + 836007, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 27 x^{14} - 116 x^{13} + 364 x^{12} - 1846 x^{11} + 5655 x^{10} - 16926 x^{9} + 57538 x^{8} - 130078 x^{7} + 296361 x^{6} - 680316 x^{5} + 1106547 x^{4} - 1586595 x^{3} + 2213979 x^{2} - 2046301 x + 836007 \)

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:  \(742561221860627884726497942833=13^{15}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{2296736265481044644940666782968803589947} a^{15} - \frac{65814099197577740575420609078182564368}{2296736265481044644940666782968803589947} a^{14} - \frac{53819127873304933215527293868539227724}{2296736265481044644940666782968803589947} a^{13} + \frac{98555335145936878829239830768674993057}{765578755160348214980222260989601196649} a^{12} + \frac{32159708399420270549622879919180719481}{2296736265481044644940666782968803589947} a^{11} - \frac{5392296439332404474754029509061618751}{255192918386782738326740753663200398883} a^{10} - \frac{994844056040333970397326980949801444719}{2296736265481044644940666782968803589947} a^{9} - \frac{521526052956104148731715825714642768496}{2296736265481044644940666782968803589947} a^{8} + \frac{952634302547705307966221477545393149944}{2296736265481044644940666782968803589947} a^{7} + \frac{1118935472092990176047512178571409946743}{2296736265481044644940666782968803589947} a^{6} - \frac{354422253649386930904100764814477556359}{765578755160348214980222260989601196649} a^{5} + \frac{241808379296997585315777418112093606917}{2296736265481044644940666782968803589947} a^{4} + \frac{751316751080113039713286151109539150414}{2296736265481044644940666782968803589947} a^{3} + \frac{899060122444071658622646915764277572257}{2296736265481044644940666782968803589947} a^{2} + \frac{193823151360924993214330970707632807530}{765578755160348214980222260989601196649} a + \frac{85996933213928174436857579673099433031}{255192918386782738326740753663200398883}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 72578331.2472 \) (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, 8.0.1530373581113.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $16$

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$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$