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Properties

Label 16.0.14754410872...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{11}\cdot 11^{5}\cdot 31^{5}$
Root discriminant $37.42$
Ramified primes $2, 5, 11, 31$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1782

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3875, -15500, 23750, -17000, 7375, -5000, 3250, 0, -101, 58, 105, 116, 35, -6, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 6*x^13 + 35*x^12 + 116*x^11 + 105*x^10 + 58*x^9 - 101*x^8 + 3250*x^6 - 5000*x^5 + 7375*x^4 - 17000*x^3 + 23750*x^2 - 15500*x + 3875)
 
gp: K = bnfinit(x^16 + 3*x^14 - 6*x^13 + 35*x^12 + 116*x^11 + 105*x^10 + 58*x^9 - 101*x^8 + 3250*x^6 - 5000*x^5 + 7375*x^4 - 17000*x^3 + 23750*x^2 - 15500*x + 3875, 1)
 

Normalized defining polynomial

\( x^{16} + 3 x^{14} - 6 x^{13} + 35 x^{12} + 116 x^{11} + 105 x^{10} + 58 x^{9} - 101 x^{8} + 3250 x^{6} - 5000 x^{5} + 7375 x^{4} - 17000 x^{3} + 23750 x^{2} - 15500 x + 3875 \)

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:  \(14754410872643200000000000=2^{16}\cdot 5^{11}\cdot 11^{5}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.42$
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:  $2, 5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{8} + \frac{2}{25} a^{7} + \frac{2}{25} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{9} + \frac{2}{25} a^{8} + \frac{2}{25} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{9} + \frac{1}{25} a^{8} + \frac{2}{25} a^{7} + \frac{2}{25} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{25} a^{13} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} - \frac{2}{25} a^{7} + \frac{1}{25} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{1375} a^{14} + \frac{12}{1375} a^{13} - \frac{1}{275} a^{12} - \frac{4}{275} a^{11} + \frac{1}{275} a^{10} + \frac{26}{1375} a^{9} + \frac{97}{1375} a^{8} + \frac{26}{275} a^{7} + \frac{1}{55} a^{6} + \frac{18}{55} a^{5} - \frac{4}{55} a^{4} - \frac{3}{55} a^{3} + \frac{4}{11} a^{2} + \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{1136182181867990125} a^{15} + \frac{55179539137478}{1136182181867990125} a^{14} + \frac{9884861915630737}{1136182181867990125} a^{13} - \frac{237036782728308}{20657857852145275} a^{12} - \frac{3440906143936376}{227236436373598025} a^{11} - \frac{14526275959697254}{1136182181867990125} a^{10} - \frac{100231359560589352}{1136182181867990125} a^{9} - \frac{24112086998168328}{1136182181867990125} a^{8} + \frac{215424077949282}{4131571570429055} a^{7} - \frac{447287080572701}{9089457454943921} a^{6} + \frac{442516195984259}{45447287274719605} a^{5} - \frac{2993375983719587}{9089457454943921} a^{4} + \frac{8884475280197307}{45447287274719605} a^{3} - \frac{1208213766071855}{9089457454943921} a^{2} - \frac{48224232318816}{293208304998191} a + \frac{102026557586747}{293208304998191}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 448395.922378 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1782:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1782 are not computed
Character table for t16n1782 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8525.1, 8.6.93024800000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31Data not computed