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Properties

Label 16.8.43988335962...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 53^{2}\cdot 283^{4}$
Root discriminant $22.53$
Ramified primes $5, 53, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1046

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 1146, -901, -1655, 1156, 1653, -275, -1603, 17, 986, -105, -319, 66, 50, -14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 14*x^14 + 50*x^13 + 66*x^12 - 319*x^11 - 105*x^10 + 986*x^9 + 17*x^8 - 1603*x^7 - 275*x^6 + 1653*x^5 + 1156*x^4 - 1655*x^3 - 901*x^2 + 1146*x - 199)
 
gp: K = bnfinit(x^16 - 3*x^15 - 14*x^14 + 50*x^13 + 66*x^12 - 319*x^11 - 105*x^10 + 986*x^9 + 17*x^8 - 1603*x^7 - 275*x^6 + 1653*x^5 + 1156*x^4 - 1655*x^3 - 901*x^2 + 1146*x - 199, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 14 x^{14} + 50 x^{13} + 66 x^{12} - 319 x^{11} - 105 x^{10} + 986 x^{9} + 17 x^{8} - 1603 x^{7} - 275 x^{6} + 1653 x^{5} + 1156 x^{4} - 1655 x^{3} - 901 x^{2} + 1146 x - 199 \)

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:  \(4398833596213134765625=5^{12}\cdot 53^{2}\cdot 283^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.53$
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:  $5, 53, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3041} a^{13} + \frac{117}{3041} a^{12} + \frac{916}{3041} a^{11} + \frac{1182}{3041} a^{10} - \frac{1267}{3041} a^{9} - \frac{900}{3041} a^{8} - \frac{710}{3041} a^{7} + \frac{505}{3041} a^{6} + \frac{797}{3041} a^{5} - \frac{1145}{3041} a^{4} + \frac{972}{3041} a^{3} + \frac{1450}{3041} a^{2} + \frac{292}{3041} a + \frac{888}{3041}$, $\frac{1}{3041} a^{14} - \frac{609}{3041} a^{12} + \frac{445}{3041} a^{11} + \frac{325}{3041} a^{10} + \frac{1371}{3041} a^{9} + \frac{1196}{3041} a^{8} + \frac{1468}{3041} a^{7} - \frac{509}{3041} a^{6} - \frac{123}{3041} a^{5} + \frac{1133}{3041} a^{4} + \frac{243}{3041} a^{3} + \frac{938}{3041} a^{2} + \frac{175}{3041} a - \frac{502}{3041}$, $\frac{1}{88189} a^{15} + \frac{3}{88189} a^{14} + \frac{4}{88189} a^{13} + \frac{24724}{88189} a^{12} + \frac{34034}{88189} a^{11} + \frac{30523}{88189} a^{10} + \frac{40585}{88189} a^{9} - \frac{32715}{88189} a^{8} + \frac{18733}{88189} a^{7} - \frac{17472}{88189} a^{6} - \frac{3317}{88189} a^{5} - \frac{38346}{88189} a^{4} + \frac{31877}{88189} a^{3} - \frac{14379}{88189} a^{2} + \frac{26969}{88189} a + \frac{41073}{88189}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 53213.8660037 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1046:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 40 conjugacy class representatives for t16n1046
Character table for t16n1046 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.283.1, 8.4.50055625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$53$53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
283Data not computed