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Properties

Label 16.0.27231268499...8081.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 43^{8}$
Root discriminant $44.89$
Ramified primes $13, 43$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22987, -32622, 212585, -177095, 35690, 21811, 13201, -11003, -6441, 3620, 2145, -1649, 222, 97, -26, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 26*x^14 + 97*x^13 + 222*x^12 - 1649*x^11 + 2145*x^10 + 3620*x^9 - 6441*x^8 - 11003*x^7 + 13201*x^6 + 21811*x^5 + 35690*x^4 - 177095*x^3 + 212585*x^2 - 32622*x + 22987)
 
gp: K = bnfinit(x^16 - 2*x^15 - 26*x^14 + 97*x^13 + 222*x^12 - 1649*x^11 + 2145*x^10 + 3620*x^9 - 6441*x^8 - 11003*x^7 + 13201*x^6 + 21811*x^5 + 35690*x^4 - 177095*x^3 + 212585*x^2 - 32622*x + 22987, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 26 x^{14} + 97 x^{13} + 222 x^{12} - 1649 x^{11} + 2145 x^{10} + 3620 x^{9} - 6441 x^{8} - 11003 x^{7} + 13201 x^{6} + 21811 x^{5} + 35690 x^{4} - 177095 x^{3} + 212585 x^{2} - 32622 x + 22987 \)

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:  \(272312684996154152285848081=13^{12}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.89$
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:  $13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{51} a^{8} + \frac{4}{51} a^{7} + \frac{19}{51} a^{6} - \frac{11}{51} a^{5} - \frac{1}{3} a^{4} + \frac{1}{51} a^{3} + \frac{1}{51} a^{2} - \frac{11}{51} a - \frac{7}{17}$, $\frac{1}{51} a^{9} + \frac{1}{17} a^{7} + \frac{5}{17} a^{6} - \frac{8}{17} a^{5} + \frac{6}{17} a^{4} - \frac{1}{17} a^{3} - \frac{5}{17} a^{2} + \frac{23}{51} a - \frac{6}{17}$, $\frac{1}{51} a^{10} + \frac{1}{17} a^{7} + \frac{7}{17} a^{6} - \frac{1}{17} a^{4} - \frac{6}{17} a^{3} + \frac{20}{51} a^{2} + \frac{5}{17} a + \frac{4}{17}$, $\frac{1}{51} a^{11} - \frac{8}{51} a^{7} - \frac{23}{51} a^{6} + \frac{13}{51} a^{5} + \frac{16}{51} a^{4} - \frac{5}{51} a^{2} - \frac{23}{51} a - \frac{5}{51}$, $\frac{1}{1989} a^{12} - \frac{8}{1989} a^{10} - \frac{16}{1989} a^{9} + \frac{11}{1989} a^{8} + \frac{304}{1989} a^{7} - \frac{10}{117} a^{6} + \frac{718}{1989} a^{5} - \frac{88}{663} a^{4} + \frac{886}{1989} a^{3} - \frac{18}{221} a^{2} - \frac{736}{1989} a - \frac{394}{1989}$, $\frac{1}{33813} a^{13} + \frac{7}{33813} a^{12} + \frac{226}{33813} a^{11} - \frac{76}{11271} a^{10} + \frac{133}{33813} a^{9} + \frac{49}{11271} a^{8} - \frac{3268}{33813} a^{7} - \frac{6751}{33813} a^{6} - \frac{15791}{33813} a^{5} + \frac{523}{2601} a^{4} + \frac{3271}{33813} a^{3} + \frac{15290}{33813} a^{2} - \frac{5975}{33813} a + \frac{8162}{33813}$, $\frac{1}{101439} a^{14} - \frac{10}{101439} a^{12} - \frac{484}{101439} a^{11} + \frac{191}{33813} a^{10} + \frac{73}{33813} a^{9} + \frac{313}{33813} a^{8} + \frac{9002}{101439} a^{7} + \frac{29443}{101439} a^{6} - \frac{20908}{101439} a^{5} + \frac{6986}{33813} a^{4} - \frac{11132}{33813} a^{3} + \frac{7457}{101439} a^{2} + \frac{2642}{101439} a + \frac{41075}{101439}$, $\frac{1}{231760632607356501631359} a^{15} - \frac{777857506843231873}{231760632607356501631359} a^{14} + \frac{3221916652941303461}{231760632607356501631359} a^{13} + \frac{16773875215067219263}{77253544202452167210453} a^{12} - \frac{1253869017099078192011}{231760632607356501631359} a^{11} + \frac{78251847318530069447}{25751181400817389070151} a^{10} + \frac{112055743998760164373}{77253544202452167210453} a^{9} + \frac{1405715814448453950710}{231760632607356501631359} a^{8} + \frac{21801396511946302051859}{231760632607356501631359} a^{7} + \frac{41185671215681777310244}{231760632607356501631359} a^{6} - \frac{71168952898474411414367}{231760632607356501631359} a^{5} - \frac{24453059639122665788885}{77253544202452167210453} a^{4} - \frac{9226070439236051336107}{231760632607356501631359} a^{3} + \frac{4344625630471881576136}{77253544202452167210453} a^{2} + \frac{5137697623393698823976}{77253544202452167210453} a + \frac{39147526681615295325667}{231760632607356501631359}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 3807723.11422 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-559}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{13}, \sqrt{-43})\), 4.2.7267.1 x2, 4.0.24037.1 x2, 8.0.97644375361.1, 8.2.383765103163.1 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$