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Properties

Label 16.0.14958734309...296.11
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}$
Root discriminant $18.24$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, 56, 48, -144, -16, 432, -160, -400, 498, -152, -124, 144, -40, -24, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 24*x^13 - 40*x^12 + 144*x^11 - 124*x^10 - 152*x^9 + 498*x^8 - 400*x^7 - 160*x^6 + 432*x^5 - 16*x^4 - 144*x^3 + 48*x^2 + 56*x + 10)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 24*x^13 - 40*x^12 + 144*x^11 - 124*x^10 - 152*x^9 + 498*x^8 - 400*x^7 - 160*x^6 + 432*x^5 - 16*x^4 - 144*x^3 + 48*x^2 + 56*x + 10, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 24 x^{13} - 40 x^{12} + 144 x^{11} - 124 x^{10} - 152 x^{9} + 498 x^{8} - 400 x^{7} - 160 x^{6} + 432 x^{5} - 16 x^{4} - 144 x^{3} + 48 x^{2} + 56 x + 10 \)

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:  \(149587343098087735296=2^{48}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{65} a^{14} - \frac{9}{65} a^{13} - \frac{9}{65} a^{12} - \frac{27}{65} a^{11} - \frac{5}{13} a^{10} + \frac{3}{65} a^{9} + \frac{1}{5} a^{8} - \frac{31}{65} a^{7} - \frac{17}{65} a^{6} + \frac{9}{65} a^{5} + \frac{5}{13} a^{4} + \frac{29}{65} a^{3} + \frac{2}{13} a^{2} - \frac{7}{65} a + \frac{5}{13}$, $\frac{1}{1159918498505} a^{15} + \frac{3954861526}{1159918498505} a^{14} - \frac{1264815672}{11957922665} a^{13} - \frac{233969980462}{1159918498505} a^{12} + \frac{2411528791}{17844899977} a^{11} + \frac{211758363613}{1159918498505} a^{10} - \frac{19235185642}{1159918498505} a^{9} + \frac{467771807589}{1159918498505} a^{8} - \frac{259196195852}{1159918498505} a^{7} - \frac{456213647131}{1159918498505} a^{6} - \frac{60757723443}{231983699701} a^{5} - \frac{47477162536}{1159918498505} a^{4} - \frac{3548866288}{17844899977} a^{3} + \frac{61496262428}{1159918498505} a^{2} + \frac{108310502275}{231983699701} a - \frac{79555017004}{231983699701}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{10665529404}{2391584533} a^{15} + \frac{89317860076}{2391584533} a^{14} - \frac{289371921460}{2391584533} a^{13} + \frac{27995782252}{183968041} a^{12} + \frac{291581869864}{2391584533} a^{11} - \frac{126660746710}{183968041} a^{10} + \frac{1937959423880}{2391584533} a^{9} + \frac{902157291657}{2391584533} a^{8} - \frac{5657160963408}{2391584533} a^{7} + \frac{6381355225672}{2391584533} a^{6} - \frac{661121108368}{2391584533} a^{5} - \frac{4385481652196}{2391584533} a^{4} + \frac{1818028598376}{2391584533} a^{3} + \frac{865992528004}{2391584533} a^{2} - \frac{839600657524}{2391584533} a - \frac{284010123271}{2391584533} \) (order $4$)
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:  \( 9194.19786468 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), 4.0.512.1, 4.0.4608.1, 4.0.27648.1, \(\Q(i, \sqrt{6})\), 4.4.27648.1, 8.0.3057647616.6, 8.0.764411904.4, 8.0.339738624.7

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 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed