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Properties

Label 16.0.25234704132...6144.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{12}\cdot 7^{2}\cdot 19^{2}$
Root discriminant $14.13$
Ramified primes $2, 3, 7, 19$
Class number $1$
Class group Trivial
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 8, 16, 8, 12, -40, 16, -84, 88, -84, 116, -84, 62, -44, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 44*x^13 + 62*x^12 - 84*x^11 + 116*x^10 - 84*x^9 + 88*x^8 - 84*x^7 + 16*x^6 - 40*x^5 + 12*x^4 + 8*x^3 + 16*x^2 + 8*x + 4)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 44*x^13 + 62*x^12 - 84*x^11 + 116*x^10 - 84*x^9 + 88*x^8 - 84*x^7 + 16*x^6 - 40*x^5 + 12*x^4 + 8*x^3 + 16*x^2 + 8*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} - 44 x^{13} + 62 x^{12} - 84 x^{11} + 116 x^{10} - 84 x^{9} + 88 x^{8} - 84 x^{7} + 16 x^{6} - 40 x^{5} + 12 x^{4} + 8 x^{3} + 16 x^{2} + 8 x + 4 \)

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:  \(2523470413267206144=2^{28}\cdot 3^{12}\cdot 7^{2}\cdot 19^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.13$
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:  $2, 3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{12} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{13} - \frac{1}{36} a^{12} + \frac{1}{18} a^{8} - \frac{2}{9} a^{7} - \frac{7}{18} a^{6} - \frac{1}{18} a^{5} - \frac{5}{18} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{2196} a^{15} + \frac{17}{2196} a^{14} - \frac{4}{549} a^{13} + \frac{5}{732} a^{12} + \frac{17}{366} a^{11} + \frac{11}{366} a^{10} + \frac{85}{1098} a^{9} + \frac{83}{1098} a^{8} - \frac{487}{1098} a^{7} + \frac{347}{1098} a^{6} - \frac{1}{549} a^{5} - \frac{5}{1098} a^{4} - \frac{85}{549} a^{3} - \frac{62}{549} a^{2} - \frac{80}{549} a + \frac{38}{183}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{67}{2196} a^{15} + \frac{22}{183} a^{14} - \frac{37}{549} a^{13} - \frac{2225}{2196} a^{12} + \frac{288}{61} a^{11} - \frac{3665}{366} a^{10} + \frac{15167}{1098} a^{9} - \frac{1085}{61} a^{8} + \frac{19331}{1098} a^{7} - \frac{6287}{549} a^{6} + \frac{448}{61} a^{5} - \frac{3935}{1098} a^{4} - \frac{466}{549} a^{3} + \frac{124}{183} a^{2} - \frac{191}{549} a - \frac{13}{549} \) (order $12$)
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:  \( 2304.27474221 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), 4.0.432.1 x2, 4.2.1728.1 x2, \(\Q(\zeta_{12})\), 8.0.2985984.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$