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Properties

Label 16.0.84467976147...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 5^{8}\cdot 11^{4}\cdot 269^{5}$
Root discriminant $55.64$
Ramified primes $2, 5, 11, 269$
Class number $176$ (GRH)
Class group $[2, 88]$ (GRH)
Galois group 16T1870

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4304, 0, 17216, 0, 27344, 0, 22460, 0, 10448, 0, 2816, 0, 427, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 33*x^14 + 427*x^12 + 2816*x^10 + 10448*x^8 + 22460*x^6 + 27344*x^4 + 17216*x^2 + 4304)
 
gp: K = bnfinit(x^16 + 33*x^14 + 427*x^12 + 2816*x^10 + 10448*x^8 + 22460*x^6 + 27344*x^4 + 17216*x^2 + 4304, 1)
 

Normalized defining polynomial

\( x^{16} + 33 x^{14} + 427 x^{12} + 2816 x^{10} + 10448 x^{8} + 22460 x^{6} + 27344 x^{4} + 17216 x^{2} + 4304 \)

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:  \(8446797614752444006400000000=2^{20}\cdot 5^{8}\cdot 11^{4}\cdot 269^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.64$
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:  $2, 5, 11, 269$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{824} a^{14} + \frac{19}{824} a^{12} - \frac{45}{824} a^{10} - \frac{7}{103} a^{8} - \frac{49}{412} a^{6} - \frac{8}{103} a^{4} + \frac{28}{103} a^{2} + \frac{9}{103}$, $\frac{1}{1648} a^{15} + \frac{19}{1648} a^{13} + \frac{161}{1648} a^{11} + \frac{75}{824} a^{9} - \frac{19}{103} a^{7} - \frac{1}{2} a^{6} + \frac{95}{206} a^{5} - \frac{1}{2} a^{4} + \frac{14}{103} a^{3} - \frac{1}{2} a^{2} - \frac{47}{103} a$

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

Class group and class number

$C_{2}\times C_{88}$, which has order $176$ (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:  \( 127879.535124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1870:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 83 conjugacy class representatives for t16n1870 are not computed
Character table for t16n1870 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.87556810000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.12.16.7$x^{12} + 5 x^{10} + 4 x^{8} + x^{6} + 4 x^{4} + x^{2} + 3$$6$$2$$16$12T42$[2, 2]_{3}^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
269Data not computed