↑

Properties

Label 16.0.10586216645...2009.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 19^{12}$
Root discriminant $23.80$
Ramified primes $3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $QD_{16}$ (as 16T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3184, -1376, -5632, 2340, 3845, -2815, -977, 2148, -251, -885, 514, 79, -151, 36, 11, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 11*x^14 + 36*x^13 - 151*x^12 + 79*x^11 + 514*x^10 - 885*x^9 - 251*x^8 + 2148*x^7 - 977*x^6 - 2815*x^5 + 3845*x^4 + 2340*x^3 - 5632*x^2 - 1376*x + 3184)
 
gp: K = bnfinit(x^16 - 7*x^15 + 11*x^14 + 36*x^13 - 151*x^12 + 79*x^11 + 514*x^10 - 885*x^9 - 251*x^8 + 2148*x^7 - 977*x^6 - 2815*x^5 + 3845*x^4 + 2340*x^3 - 5632*x^2 - 1376*x + 3184, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 11 x^{14} + 36 x^{13} - 151 x^{12} + 79 x^{11} + 514 x^{10} - 885 x^{9} - 251 x^{8} + 2148 x^{7} - 977 x^{6} - 2815 x^{5} + 3845 x^{4} + 2340 x^{3} - 5632 x^{2} - 1376 x + 3184 \)

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:  \(10586216645130957012009=3^{14}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.80$
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:  $3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{688} a^{12} - \frac{3}{344} a^{11} + \frac{7}{344} a^{10} + \frac{29}{688} a^{9} - \frac{2}{43} a^{8} + \frac{7}{86} a^{7} + \frac{67}{688} a^{6} + \frac{73}{344} a^{5} + \frac{1}{8} a^{4} + \frac{39}{688} a^{3} + \frac{7}{172} a^{2} + \frac{15}{172} a + \frac{7}{43}$, $\frac{1}{1376} a^{13} - \frac{1}{1376} a^{12} - \frac{1}{86} a^{11} + \frac{13}{1376} a^{10} + \frac{27}{1376} a^{9} - \frac{13}{172} a^{8} + \frac{3}{1376} a^{7} - \frac{207}{1376} a^{6} - \frac{27}{172} a^{5} - \frac{133}{1376} a^{4} + \frac{309}{1376} a^{3} + \frac{17}{43} a^{2} + \frac{17}{344} a - \frac{4}{43}$, $\frac{1}{5504} a^{14} - \frac{1}{2752} a^{13} + \frac{3}{5504} a^{12} + \frac{7}{5504} a^{11} + \frac{133}{2752} a^{10} + \frac{47}{5504} a^{9} - \frac{39}{5504} a^{8} + \frac{55}{2752} a^{7} - \frac{179}{5504} a^{6} + \frac{905}{5504} a^{5} + \frac{479}{2752} a^{4} - \frac{1815}{5504} a^{3} + \frac{315}{2752} a^{2} + \frac{479}{1376} a + \frac{53}{688}$, $\frac{1}{25967929456256} a^{15} + \frac{375053701}{25967929456256} a^{14} - \frac{1386747107}{25967929456256} a^{13} + \frac{3253309307}{6491982364064} a^{12} - \frac{303482001389}{25967929456256} a^{11} + \frac{702333359805}{25967929456256} a^{10} - \frac{328275928615}{12983964728128} a^{9} + \frac{1742320300821}{25967929456256} a^{8} + \frac{2715331519143}{25967929456256} a^{7} + \frac{1113048303293}{6491982364064} a^{6} - \frac{4248681254619}{25967929456256} a^{5} + \frac{1924299854867}{25967929456256} a^{4} + \frac{1903072686149}{25967929456256} a^{3} - \frac{5383914861761}{12983964728128} a^{2} - \frac{2821514853293}{6491982364064} a - \frac{510337497185}{3245991182032}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{120063}{123949564} a^{15} - \frac{20970935}{3966386048} a^{14} + \frac{2744099}{1983193024} a^{13} + \frac{177010195}{3966386048} a^{12} - \frac{355719513}{3966386048} a^{11} - \frac{188444327}{1983193024} a^{10} + \frac{1977175087}{3966386048} a^{9} - \frac{739133127}{3966386048} a^{8} - \frac{1874814317}{1983193024} a^{7} + \frac{5460261501}{3966386048} a^{6} + \frac{4670697417}{3966386048} a^{5} - \frac{4702654309}{1983193024} a^{4} + \frac{2422555561}{3966386048} a^{3} + \frac{9880328663}{1983193024} a^{2} - \frac{1113477977}{991596512} a - \frac{1767959199}{495798256} \) (order $6$)
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:  \( 452785.147047 \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-19})\), 4.2.9747.1 x2, 4.0.513.1 x2, 8.0.95004009.1, 8.2.102889341747.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$19$19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$