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Properties

Label 16.4.47009627709...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{8}\cdot 89\cdot 379^{4}$
Root discriminant $26.12$
Ramified primes $2, 5, 89, 379$
Class number $2$
Class group $[2]$
Galois group 16T1862

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, -512, -128, 32, -256, -128, -40, -75, -20, -32, -32, 2, -4, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 4*x^13 + 2*x^12 - 32*x^11 - 32*x^10 - 20*x^9 - 75*x^8 - 40*x^7 - 128*x^6 - 256*x^5 + 32*x^4 - 128*x^3 - 512*x^2 + 256)
 
gp: K = bnfinit(x^16 - 8*x^14 - 4*x^13 + 2*x^12 - 32*x^11 - 32*x^10 - 20*x^9 - 75*x^8 - 40*x^7 - 128*x^6 - 256*x^5 + 32*x^4 - 128*x^3 - 512*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 4 x^{13} + 2 x^{12} - 32 x^{11} - 32 x^{10} - 20 x^{9} - 75 x^{8} - 40 x^{7} - 128 x^{6} - 256 x^{5} + 32 x^{4} - 128 x^{3} - 512 x^{2} + 256 \)

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:  \(47009627709670400000000=2^{16}\cdot 5^{8}\cdot 89\cdot 379^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.12$
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:  $2, 5, 89, 379$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{10} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} + \frac{3}{16} a^{5} + \frac{5}{64} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{128} a^{13} - \frac{1}{32} a^{10} + \frac{1}{64} a^{9} + \frac{3}{32} a^{6} + \frac{5}{128} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a$, $\frac{1}{512} a^{14} - \frac{1}{128} a^{12} - \frac{1}{128} a^{11} - \frac{7}{256} a^{10} + \frac{1}{32} a^{9} - \frac{3}{64} a^{8} - \frac{5}{128} a^{7} + \frac{21}{512} a^{6} + \frac{9}{64} a^{5} - \frac{27}{128} a^{4} + \frac{3}{16} a^{3} - \frac{5}{32} a^{2} + \frac{3}{8}$, $\frac{1}{1024} a^{15} - \frac{1}{256} a^{13} - \frac{1}{256} a^{12} - \frac{7}{512} a^{11} - \frac{3}{64} a^{10} + \frac{5}{128} a^{9} + \frac{11}{256} a^{8} - \frac{43}{1024} a^{7} - \frac{15}{128} a^{6} + \frac{21}{256} a^{5} + \frac{1}{32} a^{4} - \frac{1}{64} a^{3} + \frac{1}{4} a^{2} + \frac{7}{16} a$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 94303.1104658 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1862:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 152 conjugacy class representatives for t16n1862 are not computed
Character table for t16n1862 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.22982560000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
379Data not computed