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Properties

Label 16.0.24073289246...0624.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 17^{4}$
Root discriminant $25.05$
Ramified primes $2, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83521, 0, -78608, 0, 41616, 0, -14960, 0, 4130, 0, -880, 0, 144, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 + 144*x^12 - 880*x^10 + 4130*x^8 - 14960*x^6 + 41616*x^4 - 78608*x^2 + 83521)
 
gp: K = bnfinit(x^16 - 16*x^14 + 144*x^12 - 880*x^10 + 4130*x^8 - 14960*x^6 + 41616*x^4 - 78608*x^2 + 83521, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} + 144 x^{12} - 880 x^{10} + 4130 x^{8} - 14960 x^{6} + 41616 x^{4} - 78608 x^{2} + 83521 \)

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:  \(24073289246567116570624=2^{58}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.05$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{136} a^{10} + \frac{1}{136} a^{8} - \frac{15}{34} a^{6} + \frac{1}{34} a^{4} - \frac{35}{136} a^{2} - \frac{3}{8}$, $\frac{1}{136} a^{11} + \frac{1}{136} a^{9} - \frac{15}{34} a^{7} + \frac{1}{34} a^{5} - \frac{35}{136} a^{3} - \frac{3}{8} a$, $\frac{1}{2312} a^{12} + \frac{1}{2312} a^{10} - \frac{16}{289} a^{8} + \frac{103}{578} a^{6} - \frac{715}{2312} a^{4} + \frac{37}{136} a^{2} - \frac{1}{2}$, $\frac{1}{2312} a^{13} + \frac{1}{2312} a^{11} - \frac{16}{289} a^{9} + \frac{103}{578} a^{7} - \frac{715}{2312} a^{5} + \frac{37}{136} a^{3} - \frac{1}{2} a$, $\frac{1}{13874312} a^{14} - \frac{1699}{13874312} a^{12} + \frac{3085}{13874312} a^{10} + \frac{311017}{6937156} a^{8} - \frac{1}{2} a^{7} + \frac{325413}{13874312} a^{6} - \frac{1}{2} a^{5} + \frac{164629}{816136} a^{4} - \frac{1}{2} a^{3} + \frac{10473}{48008} a^{2} - \frac{1}{2} a + \frac{665}{1412}$, $\frac{1}{27748624} a^{15} - \frac{1}{27748624} a^{14} - \frac{1699}{27748624} a^{13} + \frac{1699}{27748624} a^{12} + \frac{3085}{27748624} a^{11} - \frac{3085}{27748624} a^{10} - \frac{1112255}{27748624} a^{9} + \frac{1112255}{27748624} a^{8} - \frac{6611743}{27748624} a^{7} - \frac{7262569}{27748624} a^{6} - \frac{651507}{1632272} a^{5} - \frac{164629}{1632272} a^{4} - \frac{13531}{96016} a^{3} - \frac{34477}{96016} a^{2} - \frac{1847}{5648} a - \frac{977}{5648}$

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:  \( \frac{2169}{13874312} a^{14} - \frac{12519}{13874312} a^{12} + \frac{15065}{3468578} a^{10} - \frac{10637}{1734289} a^{8} - \frac{136843}{13874312} a^{6} + \frac{236837}{816136} a^{4} - \frac{5026}{6001} a^{2} + \frac{1955}{706} \) (order $16$)
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:  \( 235822.571985 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$