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Properties

Label 16.0.42998169600000000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $10.95$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, 2, 6, -20, -7, 0, 31, 0, -7, -20, 6, 2, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 5*x^14 + 2*x^13 + 6*x^12 - 20*x^11 - 7*x^10 + 31*x^8 - 7*x^6 - 20*x^5 + 6*x^4 + 2*x^3 + 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 5*x^14 + 2*x^13 + 6*x^12 - 20*x^11 - 7*x^10 + 31*x^8 - 7*x^6 - 20*x^5 + 6*x^4 + 2*x^3 + 5*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 5 x^{14} + 2 x^{13} + 6 x^{12} - 20 x^{11} - 7 x^{10} + 31 x^{8} - 7 x^{6} - 20 x^{5} + 6 x^{4} + 2 x^{3} + 5 x^{2} - 2 x + 1 \)

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:  \(42998169600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.95$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{4}{9} a^{9} - \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{9} - \frac{4}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{189} a^{14} + \frac{5}{189} a^{13} - \frac{1}{63} a^{12} + \frac{2}{21} a^{11} + \frac{1}{21} a^{10} - \frac{38}{189} a^{9} - \frac{31}{63} a^{8} - \frac{88}{189} a^{7} - \frac{17}{63} a^{6} + \frac{25}{189} a^{5} + \frac{31}{63} a^{4} - \frac{1}{63} a^{3} + \frac{3}{7} a^{2} - \frac{58}{189} a - \frac{41}{189}$, $\frac{1}{189} a^{15} - \frac{1}{27} a^{13} - \frac{1}{21} a^{12} + \frac{8}{63} a^{11} + \frac{1}{189} a^{10} - \frac{50}{189} a^{9} - \frac{22}{189} a^{8} + \frac{32}{189} a^{7} - \frac{2}{27} a^{6} + \frac{10}{189} a^{5} - \frac{16}{63} a^{4} - \frac{8}{21} a^{3} - \frac{43}{189} a^{2} + \frac{3}{7} a - \frac{89}{189}$

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:  \( -\frac{2}{9} a^{15} + \frac{50}{63} a^{14} - \frac{128}{63} a^{13} + \frac{88}{63} a^{12} - \frac{52}{63} a^{11} + \frac{233}{63} a^{10} - \frac{388}{63} a^{9} - \frac{2}{63} a^{8} + \frac{52}{63} a^{7} + \frac{530}{63} a^{6} - \frac{262}{63} a^{5} - \frac{250}{63} a^{4} - \frac{164}{63} a^{3} + \frac{214}{63} a^{2} + \frac{68}{63} a + \frac{22}{63} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 95.9988129964 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-3})\), 8.0.207360000.1, 8.4.23040000.1 x2, 8.0.3240000.1 x2, 8.0.8294400.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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$5.8.4.1$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}$