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Properties

Label 16.4.47983484309...1552.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{44}\cdot 17^{6}\cdot 113$
Root discriminant $26.16$
Ramified primes $2, 17, 113$
Class number $1$
Class group Trivial
Galois group 16T1726

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 192, -688, 1040, -1714, 2608, -2420, 1256, -321, 124, -138, 40, 15, -16, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 16*x^13 + 15*x^12 + 40*x^11 - 138*x^10 + 124*x^9 - 321*x^8 + 1256*x^7 - 2420*x^6 + 2608*x^5 - 1714*x^4 + 1040*x^3 - 688*x^2 + 192*x + 32)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 16*x^13 + 15*x^12 + 40*x^11 - 138*x^10 + 124*x^9 - 321*x^8 + 1256*x^7 - 2420*x^6 + 2608*x^5 - 1714*x^4 + 1040*x^3 - 688*x^2 + 192*x + 32, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 16 x^{13} + 15 x^{12} + 40 x^{11} - 138 x^{10} + 124 x^{9} - 321 x^{8} + 1256 x^{7} - 2420 x^{6} + 2608 x^{5} - 1714 x^{4} + 1040 x^{3} - 688 x^{2} + 192 x + 32 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(47983484309395893911552=2^{44}\cdot 17^{6}\cdot 113\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.16$
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:  $2, 17, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{12} - \frac{1}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{405232547020669393808} a^{15} + \frac{11496387132309656901}{202616273510334696904} a^{14} - \frac{3336605219700365255}{202616273510334696904} a^{13} - \frac{24421361791533790649}{101308136755167348452} a^{12} + \frac{17574139167130393743}{405232547020669393808} a^{11} + \frac{90525441510148024973}{202616273510334696904} a^{10} + \frac{13549656394270071235}{202616273510334696904} a^{9} - \frac{11521382127731076906}{25327034188791837113} a^{8} - \frac{141784850777433638441}{405232547020669393808} a^{7} - \frac{97232662350027555363}{202616273510334696904} a^{6} + \frac{839898088806718875}{5959302162068667556} a^{5} - \frac{25159384095769011571}{50654068377583674226} a^{4} - \frac{778563964025724001}{11918604324137335112} a^{3} - \frac{15112938099049637031}{101308136755167348452} a^{2} - \frac{9471512748637084689}{25327034188791837113} a + \frac{6284456442140484132}{25327034188791837113}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 498664.72805 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1726:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1726 are not computed
Character table for t16n1726 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.1212153856.4

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
113Data not computed