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Properties

Label 16.8.25620417196...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{48}\cdot 3^{14}\cdot 5^{10}\cdot 11^{7}$
Root discriminant $163.32$
Ramified primes $2, 3, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1888

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![970299, 0, 0, 0, 0, 0, -32076, 0, 3024, 0, 0, 0, 176, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 28*x^14 + 176*x^12 + 3024*x^8 - 32076*x^6 + 970299)
 
gp: K = bnfinit(x^16 - 28*x^14 + 176*x^12 + 3024*x^8 - 32076*x^6 + 970299, 1)
 

Normalized defining polynomial

\( x^{16} - 28 x^{14} + 176 x^{12} + 3024 x^{8} - 32076 x^{6} + 970299 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(256204171967704582387138560000000000=2^{48}\cdot 3^{14}\cdot 5^{10}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.32$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{8} + \frac{1}{9} a^{6} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{54} a^{10} - \frac{1}{54} a^{8} - \frac{2}{27} a^{6} + \frac{1}{3} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{594} a^{11} + \frac{5}{594} a^{9} + \frac{2}{27} a^{7} + \frac{4}{9} a^{5} + \frac{17}{66} a^{3} - \frac{1}{2} a$, $\frac{1}{1782} a^{12} + \frac{5}{1782} a^{10} + \frac{2}{81} a^{8} + \frac{4}{27} a^{6} + \frac{17}{198} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{5346} a^{13} - \frac{1}{5346} a^{11} + \frac{113}{5346} a^{9} - \frac{2}{27} a^{7} - \frac{269}{594} a^{5} - \frac{5}{22} a^{3} + \frac{1}{6} a$, $\frac{1}{222631341966} a^{14} + \frac{24109937}{222631341966} a^{12} - \frac{5757445}{1839928446} a^{10} - \frac{3480323}{204436494} a^{8} - \frac{29085013}{916178362} a^{6} - \frac{30486145}{68145498} a^{4} - \frac{1136111}{22715166} a^{2} + \frac{2974601}{7571722}$, $\frac{1}{667894025898} a^{15} + \frac{24109937}{667894025898} a^{13} + \frac{4813603}{60717638718} a^{11} - \frac{62679178}{3373202151} a^{9} - \frac{8114722247}{74210447322} a^{7} - \frac{2546047}{22715166} a^{5} - \frac{133644773}{749600478} a^{3} - \frac{135210}{3785861} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 490153091361 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147456
The 130 conjugacy class representatives for t16n1888 are not computed
Character table for t16n1888 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.198717235200000.1

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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.7.2$x^{8} - 11$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$