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Properties

Label 16.0.11698585600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 13^{4}$
Root discriminant $17.96$
Ramified primes $2, 5, 13$
Class number $3$
Class group $[3]$
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 7, -22, 52, -8, 57, -168, 389, -500, 451, -284, 158, -70, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 25*x^14 - 70*x^13 + 158*x^12 - 284*x^11 + 451*x^10 - 500*x^9 + 389*x^8 - 168*x^7 + 57*x^6 - 8*x^5 + 52*x^4 - 22*x^3 + 7*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 25*x^14 - 70*x^13 + 158*x^12 - 284*x^11 + 451*x^10 - 500*x^9 + 389*x^8 - 168*x^7 + 57*x^6 - 8*x^5 + 52*x^4 - 22*x^3 + 7*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 25 x^{14} - 70 x^{13} + 158 x^{12} - 284 x^{11} + 451 x^{10} - 500 x^{9} + 389 x^{8} - 168 x^{7} + 57 x^{6} - 8 x^{5} + 52 x^{4} - 22 x^{3} + 7 x^{2} - 2 x + 1 \)

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:  \(116985856000000000000=2^{24}\cdot 5^{12}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.96$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{2}{9} a^{12} - \frac{4}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{143870164595037} a^{15} + \frac{1189215732094}{143870164595037} a^{14} - \frac{4311249501457}{143870164595037} a^{13} + \frac{4955500513609}{143870164595037} a^{12} + \frac{21327084478652}{47956721531679} a^{11} - \frac{47842379893673}{143870164595037} a^{10} + \frac{36677013791990}{143870164595037} a^{9} - \frac{1714872001615}{47956721531679} a^{8} - \frac{2330330932819}{143870164595037} a^{7} + \frac{71900849007818}{143870164595037} a^{6} + \frac{54884013380990}{143870164595037} a^{5} + \frac{6988625460107}{47956721531679} a^{4} - \frac{55003018431335}{143870164595037} a^{3} - \frac{676711766489}{47956721531679} a^{2} + \frac{63787337197750}{143870164595037} a + \frac{17387804495342}{143870164595037}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{282715256680}{5328524614631} a^{15} - \frac{2169166893032}{5328524614631} a^{14} + \frac{28871542548140}{15985573843893} a^{13} - \frac{29968090086755}{5328524614631} a^{12} + \frac{212992205280056}{15985573843893} a^{11} - \frac{136379335786298}{5328524614631} a^{10} + \frac{660876531530344}{15985573843893} a^{9} - \frac{843936061552007}{15985573843893} a^{8} + \frac{694042242691136}{15985573843893} a^{7} - \frac{109829918363108}{5328524614631} a^{6} + \frac{22765881644864}{15985573843893} a^{5} + \frac{17698919141}{15985573843893} a^{4} + \frac{14151027229184}{5328524614631} a^{3} - \frac{121622039815238}{15985573843893} a^{2} + \frac{4237808944036}{15985573843893} a + \frac{262409724883}{15985573843893} \) (order $10$)
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:  \( 3710.59482488 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), 4.0.104000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.104000.4, 8.0.10816000000.5, 8.0.64000000.2, 8.8.432640000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$