↑

Properties

Label 16.0.37673666773...0409.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 31^{12}$
Root discriminant $34.36$
Ramified primes $3, 31$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (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![26569, -44825, 57858, -19543, -5884, 23013, -10970, 2653, 3117, -1694, 628, 165, -100, 44, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 44*x^13 - 100*x^12 + 165*x^11 + 628*x^10 - 1694*x^9 + 3117*x^8 + 2653*x^7 - 10970*x^6 + 23013*x^5 - 5884*x^4 - 19543*x^3 + 57858*x^2 - 44825*x + 26569)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 44*x^13 - 100*x^12 + 165*x^11 + 628*x^10 - 1694*x^9 + 3117*x^8 + 2653*x^7 - 10970*x^6 + 23013*x^5 - 5884*x^4 - 19543*x^3 + 57858*x^2 - 44825*x + 26569, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 44 x^{13} - 100 x^{12} + 165 x^{11} + 628 x^{10} - 1694 x^{9} + 3117 x^{8} + 2653 x^{7} - 10970 x^{6} + 23013 x^{5} - 5884 x^{4} - 19543 x^{3} + 57858 x^{2} - 44825 x + 26569 \)

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:  \(3767366677314336061820409=3^{14}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.36$
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:  $3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{42021} a^{14} - \frac{292}{6003} a^{13} + \frac{85}{42021} a^{12} - \frac{1193}{42021} a^{11} - \frac{2026}{42021} a^{10} + \frac{1621}{42021} a^{9} + \frac{2069}{14007} a^{8} + \frac{4805}{14007} a^{7} + \frac{2549}{14007} a^{6} - \frac{4226}{42021} a^{5} - \frac{13975}{42021} a^{4} - \frac{346}{1827} a^{3} + \frac{6472}{42021} a^{2} + \frac{2567}{6003} a + \frac{19483}{42021}$, $\frac{1}{1393878128769} a^{15} - \frac{1794848}{464626042923} a^{14} - \frac{7694311981}{1393878128769} a^{13} + \frac{32250087883}{199125446967} a^{12} + \frac{11135721551}{464626042923} a^{11} - \frac{173639818315}{1393878128769} a^{10} + \frac{22615085683}{464626042923} a^{9} + \frac{29470913809}{464626042923} a^{8} + \frac{19450854025}{154875347641} a^{7} + \frac{663446059231}{1393878128769} a^{6} - \frac{180855591440}{464626042923} a^{5} + \frac{362207312606}{1393878128769} a^{4} + \frac{83716230988}{199125446967} a^{3} + \frac{112975414486}{464626042923} a^{2} - \frac{643133648911}{1393878128769} a + \frac{396760481}{950155507}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( -\frac{355380}{33170989} a^{15} + \frac{4974994}{232196923} a^{14} + \frac{382690}{99512967} a^{13} - \frac{329134286}{696590769} a^{12} + \frac{724004653}{696590769} a^{11} - \frac{48980430}{232196923} a^{10} - \frac{1672260158}{232196923} a^{9} + \frac{4014291618}{232196923} a^{8} - \frac{2197305258}{232196923} a^{7} - \frac{9694545180}{232196923} a^{6} + \frac{27048519133}{232196923} a^{5} - \frac{62448949016}{696590769} a^{4} - \frac{38406665960}{696590769} a^{3} + \frac{190893569686}{696590769} a^{2} - \frac{8199919774}{33170989} a + \frac{210075442}{1424521} \) (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:  \( 38710.9719245 \) (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

\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{93}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-31})\), 4.2.25947.1 x2, 4.0.837.1 x2, 8.0.673246809.1, 8.2.1940970550347.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$31$31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$