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Properties

Label 16.8.27339088708...0864.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{4}\cdot 17^{4}\cdot 97^{2}$
Root discriminant $18.94$
Ramified primes $2, 3, 17, 97$
Class number $1$
Class group Trivial
Galois group 16T1553

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 12, 18, -40, -21, 32, -6, 44, -70, 32, 20, -56, 70, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 20*x^10 + 32*x^9 - 70*x^8 + 44*x^7 - 6*x^6 + 32*x^5 - 21*x^4 - 40*x^3 + 18*x^2 + 12*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 20*x^10 + 32*x^9 - 70*x^8 + 44*x^7 - 6*x^6 + 32*x^5 - 21*x^4 - 40*x^3 + 18*x^2 + 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 70 x^{12} - 56 x^{11} + 20 x^{10} + 32 x^{9} - 70 x^{8} + 44 x^{7} - 6 x^{6} + 32 x^{5} - 21 x^{4} - 40 x^{3} + 18 x^{2} + 12 x + 1 \)

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:  \(273390887084557860864=2^{32}\cdot 3^{4}\cdot 17^{4}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.94$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{4}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{1}{27} a^{12} + \frac{4}{27} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{2}{27} a^{8} + \frac{7}{27} a^{7} - \frac{11}{27} a^{6} + \frac{11}{27} a^{5} - \frac{2}{27} a^{4} + \frac{1}{27} a^{3} + \frac{1}{9} a^{2} + \frac{4}{27} a + \frac{7}{27}$, $\frac{1}{1917} a^{15} + \frac{28}{1917} a^{14} + \frac{14}{639} a^{13} - \frac{106}{1917} a^{12} - \frac{196}{1917} a^{11} - \frac{25}{213} a^{10} - \frac{838}{1917} a^{9} - \frac{529}{1917} a^{8} + \frac{208}{639} a^{7} + \frac{214}{1917} a^{6} + \frac{740}{1917} a^{5} + \frac{134}{639} a^{4} - \frac{814}{1917} a^{3} - \frac{305}{1917} a^{2} + \frac{251}{639} a + \frac{779}{1917}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15487.1799506 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1553:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 106 conjugacy class representatives for t16n1553 are not computed
Character table for t16n1553 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.4.1837170688.2

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$