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Properties

Label 16.8.35465739761...2784.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 17^{10}$
Root discriminant $39.52$
Ramified primes $2, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 0, 0, 0, -2278, 0, 1496, 0, 1426, 0, -80, 0, -134, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 134*x^12 - 80*x^10 + 1426*x^8 + 1496*x^6 - 2278*x^4 + 289)
 
gp: K = bnfinit(x^16 - 8*x^14 - 134*x^12 - 80*x^10 + 1426*x^8 + 1496*x^6 - 2278*x^4 + 289, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 134 x^{12} - 80 x^{10} + 1426 x^{8} + 1496 x^{6} - 2278 x^{4} + 289 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35465739761106676596342784=2^{44}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.52$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{952} a^{12} - \frac{53}{238} a^{10} + \frac{53}{952} a^{8} - \frac{27}{119} a^{6} + \frac{457}{952} a^{4} - \frac{3}{14} a^{2} + \frac{13}{56}$, $\frac{1}{952} a^{13} - \frac{53}{238} a^{11} + \frac{53}{952} a^{9} - \frac{27}{119} a^{7} + \frac{457}{952} a^{5} - \frac{3}{14} a^{3} + \frac{13}{56} a$, $\frac{1}{8314728016} a^{14} + \frac{1163187}{8314728016} a^{12} - \frac{1}{4} a^{11} + \frac{498768325}{8314728016} a^{10} - \frac{1}{4} a^{9} + \frac{202695399}{8314728016} a^{8} - \frac{153056847}{8314728016} a^{6} - \frac{3147090957}{8314728016} a^{4} - \frac{1}{4} a^{3} - \frac{231840059}{489101648} a^{2} - \frac{1}{4} a - \frac{130102377}{489101648}$, $\frac{1}{8314728016} a^{15} + \frac{1163187}{8314728016} a^{13} + \frac{498768325}{8314728016} a^{11} - \frac{1}{4} a^{10} + \frac{202695399}{8314728016} a^{9} - \frac{1}{4} a^{8} - \frac{153056847}{8314728016} a^{7} - \frac{3147090957}{8314728016} a^{5} - \frac{231840059}{489101648} a^{3} - \frac{1}{4} a^{2} - \frac{130102377}{489101648} a - \frac{1}{4}$

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:  $11$
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:  \( 8015770.40158 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.24.30$x^{8} + 16 x^{7} + 16$$8$$1$$24$$Q_8:C_2$$[2, 3, 4]^{2}$
$17$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$