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Properties

Label 16.8.33634732739...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{36}\cdot 5^{8}\cdot 11^{6}\cdot 29^{4}$
Root discriminant $60.66$
Ramified primes $2, 5, 11, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T511)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2621, -9868, 58120, -24820, -18547, 12564, -6310, -160, -1083, 1556, -130, 60, 23, -8, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2*x^14 - 8*x^13 + 23*x^12 + 60*x^11 - 130*x^10 + 1556*x^9 - 1083*x^8 - 160*x^7 - 6310*x^6 + 12564*x^5 - 18547*x^4 - 24820*x^3 + 58120*x^2 - 9868*x - 2621)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2*x^14 - 8*x^13 + 23*x^12 + 60*x^11 - 130*x^10 + 1556*x^9 - 1083*x^8 - 160*x^7 - 6310*x^6 + 12564*x^5 - 18547*x^4 - 24820*x^3 + 58120*x^2 - 9868*x - 2621, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2 x^{14} - 8 x^{13} + 23 x^{12} + 60 x^{11} - 130 x^{10} + 1556 x^{9} - 1083 x^{8} - 160 x^{7} - 6310 x^{6} + 12564 x^{5} - 18547 x^{4} - 24820 x^{3} + 58120 x^{2} - 9868 x - 2621 \)

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:  \(33634732739038648729600000000=2^{36}\cdot 5^{8}\cdot 11^{6}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.66$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1279} a^{14} - \frac{603}{1279} a^{13} + \frac{295}{1279} a^{12} + \frac{639}{1279} a^{11} - \frac{578}{1279} a^{10} - \frac{216}{1279} a^{9} + \frac{491}{1279} a^{8} - \frac{312}{1279} a^{7} + \frac{63}{1279} a^{6} - \frac{608}{1279} a^{5} - \frac{155}{1279} a^{4} - \frac{67}{1279} a^{3} - \frac{280}{1279} a^{2} + \frac{457}{1279} a + \frac{17}{1279}$, $\frac{1}{5310625580379211656421421777599897} a^{15} + \frac{1593713373878828348092673239601}{5310625580379211656421421777599897} a^{14} + \frac{1455783782941229132445069174069927}{5310625580379211656421421777599897} a^{13} + \frac{2060337353117389518147467249797334}{5310625580379211656421421777599897} a^{12} - \frac{414392146856745784641544125996644}{5310625580379211656421421777599897} a^{11} + \frac{123204449526010429829537793582835}{758660797197030236631631682514271} a^{10} - \frac{1223922064407689885701355484599067}{5310625580379211656421421777599897} a^{9} + \frac{817220779245351260444671122641992}{5310625580379211656421421777599897} a^{8} - \frac{194606298899732686995587534932227}{758660797197030236631631682514271} a^{7} - \frac{172999058046431748333676553883208}{5310625580379211656421421777599897} a^{6} + \frac{398045406866877500860883562436143}{5310625580379211656421421777599897} a^{5} + \frac{607285893849450314490128334224738}{5310625580379211656421421777599897} a^{4} + \frac{121855300633175186465191893678224}{758660797197030236631631682514271} a^{3} + \frac{306557571879108932636822269443947}{5310625580379211656421421777599897} a^{2} + \frac{2278265863237780728890668959224210}{5310625580379211656421421777599897} a - \frac{598603799119727256997107775208495}{5310625580379211656421421777599897}$

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:  \( 129329935.569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T511):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.4.127600.2, 4.4.725.1, 8.4.218071040000.8, 8.4.183397744640000.2, 8.8.16281760000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$