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Properties

Label 16.0.25076252558...7241.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 19^{14}$
Root discriminant $79.42$
Ramified primes $11, 19$
Class number $1$ (GRH)
Class group Trivial (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![1430416, 1365832, 407384, -522598, 65135, 180066, 96471, 56768, 24209, 7938, 2668, 288, 79, 92, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 5*x^14 + 92*x^13 + 79*x^12 + 288*x^11 + 2668*x^10 + 7938*x^9 + 24209*x^8 + 56768*x^7 + 96471*x^6 + 180066*x^5 + 65135*x^4 - 522598*x^3 + 407384*x^2 + 1365832*x + 1430416)
 
gp: K = bnfinit(x^16 - 2*x^15 + 5*x^14 + 92*x^13 + 79*x^12 + 288*x^11 + 2668*x^10 + 7938*x^9 + 24209*x^8 + 56768*x^7 + 96471*x^6 + 180066*x^5 + 65135*x^4 - 522598*x^3 + 407384*x^2 + 1365832*x + 1430416, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 5 x^{14} + 92 x^{13} + 79 x^{12} + 288 x^{11} + 2668 x^{10} + 7938 x^{9} + 24209 x^{8} + 56768 x^{7} + 96471 x^{6} + 180066 x^{5} + 65135 x^{4} - 522598 x^{3} + 407384 x^{2} + 1365832 x + 1430416 \)

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:  \(2507625255850803120915676947241=11^{12}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.42$
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:  $11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{368} a^{10} + \frac{3}{368} a^{9} - \frac{1}{184} a^{8} + \frac{3}{184} a^{7} - \frac{9}{92} a^{6} + \frac{37}{184} a^{5} + \frac{49}{368} a^{4} - \frac{71}{368} a^{3} + \frac{5}{46} a^{2} - \frac{4}{23} a - \frac{1}{2}$, $\frac{1}{736} a^{11} + \frac{35}{736} a^{9} - \frac{17}{368} a^{8} - \frac{1}{92} a^{7} - \frac{1}{368} a^{6} + \frac{149}{736} a^{5} + \frac{13}{92} a^{4} - \frac{15}{32} a^{3} + \frac{1}{8} a^{2} - \frac{11}{46} a - \frac{1}{4}$, $\frac{1}{8096} a^{12} - \frac{1}{2024} a^{11} - \frac{5}{8096} a^{10} - \frac{147}{4048} a^{9} + \frac{3}{1012} a^{8} - \frac{105}{4048} a^{7} - \frac{243}{8096} a^{6} - \frac{311}{2024} a^{5} - \frac{1985}{8096} a^{4} + \frac{401}{1012} a^{3} + \frac{215}{1012} a^{2} + \frac{35}{92} a - \frac{1}{11}$, $\frac{1}{32384} a^{13} + \frac{1}{32384} a^{12} - \frac{3}{32384} a^{11} + \frac{3}{2944} a^{10} - \frac{411}{8096} a^{9} - \frac{265}{16192} a^{8} + \frac{2667}{32384} a^{7} - \frac{4043}{32384} a^{6} + \frac{5941}{32384} a^{5} - \frac{5397}{32384} a^{4} - \frac{5893}{16192} a^{3} - \frac{29}{88} a^{2} + \frac{1569}{4048} a - \frac{43}{88}$, $\frac{1}{1683968} a^{14} - \frac{9}{841984} a^{13} - \frac{3}{841984} a^{12} + \frac{497}{841984} a^{11} - \frac{2175}{1683968} a^{10} - \frac{16347}{841984} a^{9} - \frac{2795}{129536} a^{8} + \frac{45541}{420992} a^{7} - \frac{8383}{76544} a^{6} + \frac{29057}{420992} a^{5} - \frac{11031}{129536} a^{4} - \frac{279765}{841984} a^{3} - \frac{52761}{210496} a^{2} - \frac{50357}{210496} a - \frac{119}{352}$, $\frac{1}{2285522430619451187946496} a^{15} + \frac{545980724905552711}{2285522430619451187946496} a^{14} - \frac{248634743332528183}{35711287978428924811664} a^{13} + \frac{1812498750436585571}{51943691604987526998784} a^{12} + \frac{1214071828970610364907}{2285522430619451187946496} a^{11} + \frac{2047439181075368251003}{2285522430619451187946496} a^{10} + \frac{1019452430557504770183}{175809417739957783688192} a^{9} - \frac{121636241301725446944627}{2285522430619451187946496} a^{8} + \frac{102712170672657521203809}{1142761215309725593973248} a^{7} - \frac{3039478645387759172777}{49685270230857634520576} a^{6} + \frac{13845904733631738571541}{175809417739957783688192} a^{5} + \frac{368889352557946896349963}{2285522430619451187946496} a^{4} + \frac{564881248926394576239087}{1142761215309725593973248} a^{3} - \frac{52466531271232653357527}{142845151913715699246656} a^{2} - \frac{39676617049676932621}{955485965978031433088} a + \frac{682286245786865491}{1597802618692360256}$

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:  \( -1 \) (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:  \( 4701685160.8 \) (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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{209}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-11}, \sqrt{-19})\), 4.2.829939.1 x2, 4.0.75449.1 x2, 8.0.688798743721.1, 8.2.1583548311814579.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
19Data not computed