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Properties

Label 16.0.13319580214...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 2729^{3}$
Root discriminant $27.88$
Ramified primes $2, 5, 2729$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2.D_4^2.C_2$ (as 16T660)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431, -768, 1076, -1690, 1938, -2090, 1565, -924, 401, 40, -37, 90, -3, 4, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 + 4*x^13 - 3*x^12 + 90*x^11 - 37*x^10 + 40*x^9 + 401*x^8 - 924*x^7 + 1565*x^6 - 2090*x^5 + 1938*x^4 - 1690*x^3 + 1076*x^2 - 768*x + 431)
 
gp: K = bnfinit(x^16 - 2*x^15 + 9*x^14 + 4*x^13 - 3*x^12 + 90*x^11 - 37*x^10 + 40*x^9 + 401*x^8 - 924*x^7 + 1565*x^6 - 2090*x^5 + 1938*x^4 - 1690*x^3 + 1076*x^2 - 768*x + 431, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 9 x^{14} + 4 x^{13} - 3 x^{12} + 90 x^{11} - 37 x^{10} + 40 x^{9} + 401 x^{8} - 924 x^{7} + 1565 x^{6} - 2090 x^{5} + 1938 x^{4} - 1690 x^{3} + 1076 x^{2} - 768 x + 431 \)

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:  \(133195802142310400000000=2^{24}\cdot 5^{8}\cdot 2729^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.88$
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:  $2, 5, 2729$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{1}{11} a^{9} + \frac{4}{11} a^{8} + \frac{2}{11} a^{7} + \frac{3}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a^{2} - \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{3}{11} a^{9} - \frac{2}{11} a^{8} + \frac{1}{11} a^{7} - \frac{1}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} - \frac{2}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{13} + \frac{3}{11} a^{10} - \frac{2}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{2}{11} a^{6} + \frac{4}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{121} a^{14} + \frac{3}{121} a^{13} + \frac{4}{121} a^{12} - \frac{3}{121} a^{11} - \frac{10}{121} a^{10} + \frac{56}{121} a^{9} + \frac{14}{121} a^{8} - \frac{9}{121} a^{7} - \frac{1}{121} a^{6} - \frac{56}{121} a^{5} - \frac{59}{121} a^{4} + \frac{4}{11} a^{3} - \frac{50}{121} a^{2} - \frac{37}{121} a + \frac{10}{121}$, $\frac{1}{4287105069661667} a^{15} - \frac{11249398306734}{4287105069661667} a^{14} - \frac{93315976065992}{4287105069661667} a^{13} + \frac{183258291704860}{4287105069661667} a^{12} + \frac{4995766703777}{4287105069661667} a^{11} + \frac{1643989502790408}{4287105069661667} a^{10} - \frac{324186022984268}{4287105069661667} a^{9} + \frac{13570854208311}{389736824514697} a^{8} - \frac{189118111987778}{4287105069661667} a^{7} - \frac{2083415308197029}{4287105069661667} a^{6} - \frac{446429923788086}{4287105069661667} a^{5} + \frac{409503430568231}{4287105069661667} a^{4} - \frac{1147613225509339}{4287105069661667} a^{3} - \frac{99144605721918}{389736824514697} a^{2} + \frac{454890994634283}{4287105069661667} a + \frac{20055078091166}{4287105069661667}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 19256.9486953 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T660):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.6986240000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
2729Data not computed