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Properties

Label 16.0.93492089436...3456.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{12}$
Root discriminant $15.33$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -24, 96, -204, 232, -76, -88, 192, -125, 4, 52, -56, 28, -12, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 12*x^13 + 28*x^12 - 56*x^11 + 52*x^10 + 4*x^9 - 125*x^8 + 192*x^7 - 88*x^6 - 76*x^5 + 232*x^4 - 204*x^3 + 96*x^2 - 24*x + 3)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 12*x^13 + 28*x^12 - 56*x^11 + 52*x^10 + 4*x^9 - 125*x^8 + 192*x^7 - 88*x^6 - 76*x^5 + 232*x^4 - 204*x^3 + 96*x^2 - 24*x + 3, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 28 x^{12} - 56 x^{11} + 52 x^{10} + 4 x^{9} - 125 x^{8} + 192 x^{7} - 88 x^{6} - 76 x^{5} + 232 x^{4} - 204 x^{3} + 96 x^{2} - 24 x + 3 \)

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:  \(9349208943630483456=2^{44}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.33$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{9} - \frac{4}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{78964745517} a^{15} + \frac{59251028}{11280677931} a^{14} + \frac{10881348850}{78964745517} a^{13} + \frac{3997767710}{78964745517} a^{12} + \frac{4491281434}{78964745517} a^{11} - \frac{2534191270}{78964745517} a^{10} - \frac{28922871199}{78964745517} a^{9} - \frac{13654250906}{78964745517} a^{8} + \frac{2601201584}{11280677931} a^{7} + \frac{4526744617}{78964745517} a^{6} + \frac{20279736968}{78964745517} a^{5} - \frac{34420569962}{78964745517} a^{4} + \frac{631864150}{3760225977} a^{3} + \frac{1763220706}{26321581839} a^{2} - \frac{11204390654}{26321581839} a + \frac{5184407548}{26321581839}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{392956}{22606569} a^{15} + \frac{329272}{2511841} a^{14} - \frac{12043108}{22606569} a^{13} + \frac{21568988}{22606569} a^{12} - \frac{8630320}{7535523} a^{11} + \frac{79832020}{22606569} a^{10} - \frac{164329036}{22606569} a^{9} + \frac{12465953}{2511841} a^{8} + \frac{48556504}{22606569} a^{7} - \frac{440119928}{22606569} a^{6} + \frac{173874716}{7535523} a^{5} - \frac{55391200}{22606569} a^{4} - \frac{105718124}{7535523} a^{3} + \frac{246349784}{7535523} a^{2} - \frac{123667328}{7535523} a + \frac{31891286}{7535523} \) (order $6$)
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:  \( 7617.57294068 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), 4.0.432.1, 4.0.1728.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.0.47775744.3, 8.0.191102976.1 x2, 8.0.254803968.2 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$