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Properties

Label 16.8.37319869539...8841.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 53^{6}$
Root discriminant $52.87$
Ramified primes $17, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, -881664, 0, 313296, 0, -80581, 0, 23808, 0, -5908, 0, 804, 0, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 804*x^12 - 5908*x^10 + 23808*x^8 - 80581*x^6 + 313296*x^4 - 881664*x^2 + 1048576)
 
gp: K = bnfinit(x^16 - 48*x^14 + 804*x^12 - 5908*x^10 + 23808*x^8 - 80581*x^6 + 313296*x^4 - 881664*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{14} + 804 x^{12} - 5908 x^{10} + 23808 x^{8} - 80581 x^{6} + 313296 x^{4} - 881664 x^{2} + 1048576 \)

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:  \(3731986953978689760254088841=17^{14}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.87$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{8} + \frac{1}{10} a^{6} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{7} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a$, $\frac{1}{30} a^{10} + \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{2}{15}$, $\frac{1}{60} a^{11} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{11}{60} a$, $\frac{1}{1440} a^{12} - \frac{1}{120} a^{11} - \frac{1}{90} a^{10} + \frac{1}{40} a^{8} + \frac{1}{10} a^{7} + \frac{29}{120} a^{6} - \frac{2}{5} a^{5} + \frac{1}{15} a^{4} + \frac{2}{5} a^{3} + \frac{31}{288} a^{2} + \frac{49}{120} a - \frac{7}{45}$, $\frac{1}{5760} a^{13} - \frac{1}{360} a^{11} + \frac{1}{160} a^{9} + \frac{89}{480} a^{7} + \frac{4}{15} a^{5} + \frac{319}{1152} a^{3} - \frac{1}{2} a^{2} - \frac{149}{360} a$, $\frac{1}{4677508823040} a^{14} - \frac{1}{11520} a^{13} + \frac{85521577}{292344301440} a^{12} - \frac{1}{144} a^{11} + \frac{373305289}{1169377205760} a^{10} - \frac{1}{320} a^{9} - \frac{5705198167}{389792401920} a^{8} - \frac{233}{960} a^{7} - \frac{177917657}{1015084380} a^{6} + \frac{7}{15} a^{5} - \frac{1646675073733}{4677508823040} a^{4} + \frac{3013}{11520} a^{3} + \frac{15875604257}{292344301440} a^{2} - \frac{277}{720} a - \frac{427148228}{2283939855}$, $\frac{1}{149680282337280} a^{15} + \frac{138914441}{1871003529216} a^{13} + \frac{195269506249}{37420070584320} a^{11} - \frac{54429248407}{12473356861440} a^{9} + \frac{2943188921}{64965400320} a^{7} - \frac{57776780950213}{149680282337280} a^{5} - \frac{1}{2} a^{4} - \frac{4274886070003}{9355017646080} a^{3} - \frac{1}{2} a^{2} + \frac{13928662747}{29234430144} a - \frac{1}{2}$

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:  \( 109824803.846 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.260389.1, 4.4.4913.1, 4.4.15317.1, 8.4.61089990620221.2, 8.4.61089990620221.1, 8.8.67802431321.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
17Data not computed
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$