↑

Properties

Label 16.0.94041827573...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{8}\cdot 11^{6}$
Root discriminant $20.46$
Ramified primes $2, 3, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_4^2.C_2$ (as 16T602)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 216, 324, 252, 131, 370, 498, 328, -91, -44, 152, -112, 31, -8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 8*x^13 + 31*x^12 - 112*x^11 + 152*x^10 - 44*x^9 - 91*x^8 + 328*x^7 + 498*x^6 + 370*x^5 + 131*x^4 + 252*x^3 + 324*x^2 + 216*x + 81)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 8*x^13 + 31*x^12 - 112*x^11 + 152*x^10 - 44*x^9 - 91*x^8 + 328*x^7 + 498*x^6 + 370*x^5 + 131*x^4 + 252*x^3 + 324*x^2 + 216*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 8 x^{13} + 31 x^{12} - 112 x^{11} + 152 x^{10} - 44 x^{9} - 91 x^{8} + 328 x^{7} + 498 x^{6} + 370 x^{5} + 131 x^{4} + 252 x^{3} + 324 x^{2} + 216 x + 81 \)

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:  \(940418275737600000000=2^{24}\cdot 3^{4}\cdot 5^{8}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.46$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{171} a^{14} - \frac{8}{171} a^{13} - \frac{20}{171} a^{12} + \frac{4}{171} a^{11} + \frac{79}{171} a^{10} - \frac{1}{9} a^{9} + \frac{50}{171} a^{8} - \frac{2}{171} a^{7} + \frac{83}{171} a^{6} - \frac{26}{171} a^{5} - \frac{7}{57} a^{4} - \frac{26}{171} a^{3} - \frac{82}{171} a^{2} + \frac{17}{57} a - \frac{7}{19}$, $\frac{1}{14704451639171668863} a^{15} - \frac{29592371134608860}{14704451639171668863} a^{14} - \frac{767864403614753366}{14704451639171668863} a^{13} - \frac{15486553072806872}{241056584248715883} a^{12} + \frac{3132865543176660793}{14704451639171668863} a^{11} + \frac{1247192675309473484}{14704451639171668863} a^{10} - \frac{2553785790104402659}{14704451639171668863} a^{9} - \frac{7097763788925236099}{14704451639171668863} a^{8} - \frac{2808303649598135683}{14704451639171668863} a^{7} - \frac{1895286804719382140}{14704451639171668863} a^{6} + \frac{26421099234311650}{80352194749571961} a^{5} + \frac{6889128479079237856}{14704451639171668863} a^{4} - \frac{72706956116984059}{14704451639171668863} a^{3} + \frac{221944102425126380}{1633827959907963207} a^{2} + \frac{666804666468454586}{1633827959907963207} a + \frac{216186361660728908}{544609319969321069}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2165.53957155 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4^2.C_2$ (as 16T602):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$
Character table for $C_2\times D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 4.4.4400.1, 4.2.400.1, 8.0.1916640000.1, 8.0.1916640000.2, 8.4.19360000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$