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Properties

Label 16.0.41469265383...0448.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{59}\cdot 449^{2}\cdot 1889^{2}$
Root discriminant $70.98$
Ramified primes $2, 449, 1889$
Class number $7128$ (GRH)
Class group $[2, 2, 1782]$ (GRH)
Galois group 16T1584

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1612808, 0, 5617888, 0, 4988280, 0, 1996208, 0, 418966, 0, 47936, 0, 2928, 0, 88, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 88*x^14 + 2928*x^12 + 47936*x^10 + 418966*x^8 + 1996208*x^6 + 4988280*x^4 + 5617888*x^2 + 1612808)
 
gp: K = bnfinit(x^16 + 88*x^14 + 2928*x^12 + 47936*x^10 + 418966*x^8 + 1996208*x^6 + 4988280*x^4 + 5617888*x^2 + 1612808, 1)
 

Normalized defining polynomial

\( x^{16} + 88 x^{14} + 2928 x^{12} + 47936 x^{10} + 418966 x^{8} + 1996208 x^{6} + 4988280 x^{4} + 5617888 x^{2} + 1612808 \)

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:  \(414692653834021167975731560448=2^{59}\cdot 449^{2}\cdot 1889^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.98$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{8} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{9} - \frac{3}{8} a^{5} + \frac{1}{4} a$, $\frac{1}{9089479465857394192} a^{14} - \frac{32899046880613075}{1136184933232174274} a^{12} + \frac{954803642717463091}{4544739732928697096} a^{10} + \frac{16646343928100549}{568092466616087137} a^{8} + \frac{39255359282397093}{267337631348746888} a^{6} + \frac{1114378318563719}{33417203918593361} a^{4} - \frac{1005577549040589527}{2272369866464348548} a^{2} - \frac{621525644649394}{1265239346583713}$, $\frac{1}{9089479465857394192} a^{15} - \frac{32899046880613075}{1136184933232174274} a^{13} + \frac{954803642717463091}{4544739732928697096} a^{11} + \frac{16646343928100549}{568092466616087137} a^{9} + \frac{39255359282397093}{267337631348746888} a^{7} + \frac{1114378318563719}{33417203918593361} a^{5} - \frac{1005577549040589527}{2272369866464348548} a^{3} - \frac{621525644649394}{1265239346583713} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1782}$, which has order $7128$ (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:  \( 56271.9156358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1584:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1584 are not computed
Character table for t16n1584 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.31.33$x^{8} + 12 x^{4} + 34$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
2.8.28.25$x^{8} + 8 x^{5} + 8 x^{4} + 30$$8$$1$$28$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
449Data not computed
1889Data not computed