↑

Properties

Label 16.0.15492243194...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{4}\cdot 7^{8}$
Root discriminant $13.71$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $Q_8 : C_2^2$ (as 16T23)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -28, 90, -114, 55, -30, 105, -68, -49, 46, 28, -4, -35, 12, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 + 12*x^13 - 35*x^12 - 4*x^11 + 28*x^10 + 46*x^9 - 49*x^8 - 68*x^7 + 105*x^6 - 30*x^5 + 55*x^4 - 114*x^3 + 90*x^2 - 28*x + 4)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 + 12*x^13 - 35*x^12 - 4*x^11 + 28*x^10 + 46*x^9 - 49*x^8 - 68*x^7 + 105*x^6 - 30*x^5 + 55*x^4 - 114*x^3 + 90*x^2 - 28*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} + 12 x^{13} - 35 x^{12} - 4 x^{11} + 28 x^{10} + 46 x^{9} - 49 x^{8} - 68 x^{7} + 105 x^{6} - 30 x^{5} + 55 x^{4} - 114 x^{3} + 90 x^{2} - 28 x + 4 \)

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:  \(1549224319426560000=2^{16}\cdot 3^{8}\cdot 5^{4}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.71$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4}$, $\frac{1}{40} a^{12} + \frac{3}{40} a^{11} - \frac{1}{20} a^{10} + \frac{3}{40} a^{9} - \frac{1}{20} a^{8} + \frac{1}{40} a^{7} + \frac{1}{8} a^{5} + \frac{17}{40} a^{4} - \frac{1}{10} a^{3} - \frac{1}{20} a^{2} - \frac{1}{10}$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{1}{40} a^{9} + \frac{7}{40} a^{8} + \frac{7}{40} a^{7} - \frac{1}{8} a^{6} + \frac{3}{10} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{7}{20} a^{2} - \frac{1}{10} a + \frac{3}{10}$, $\frac{1}{320} a^{14} + \frac{1}{160} a^{13} - \frac{1}{80} a^{12} + \frac{9}{160} a^{11} - \frac{7}{320} a^{10} - \frac{7}{160} a^{9} + \frac{7}{320} a^{8} + \frac{9}{80} a^{7} - \frac{7}{80} a^{6} + \frac{9}{20} a^{5} - \frac{11}{320} a^{4} + \frac{9}{160} a^{3} + \frac{7}{160} a^{2} + \frac{21}{80} a + \frac{29}{80}$, $\frac{1}{188480} a^{15} - \frac{49}{37696} a^{14} - \frac{757}{94240} a^{13} + \frac{139}{18848} a^{12} + \frac{4231}{37696} a^{11} + \frac{2283}{188480} a^{10} + \frac{30433}{188480} a^{9} + \frac{37227}{188480} a^{8} + \frac{2217}{11780} a^{7} + \frac{109}{496} a^{6} - \frac{88891}{188480} a^{5} - \frac{43897}{188480} a^{4} - \frac{752}{2945} a^{3} + \frac{13201}{94240} a^{2} + \frac{159}{760} a + \frac{3049}{9424}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{18097}{188480} a^{15} + \frac{47301}{94240} a^{14} - \frac{20087}{47120} a^{13} - \frac{164829}{94240} a^{12} + \frac{428511}{188480} a^{11} + \frac{55597}{18848} a^{10} - \frac{303119}{188480} a^{9} - \frac{31909}{4712} a^{8} - \frac{1133}{47120} a^{7} + \frac{11657}{1240} a^{6} - \frac{98457}{37696} a^{5} - \frac{267859}{94240} a^{4} - \frac{552259}{94240} a^{3} + \frac{327141}{47120} a^{2} + \frac{121}{1520} a - \frac{12513}{11780} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3096.16674848 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{7})\), 8.0.49787136.1, 8.0.138297600.1 x2, 8.0.25401600.1 x2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$