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Properties

Label 16.0.19923381218...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{10}\cdot 41^{6}$
Root discriminant $44.03$
Ramified primes $2, 5, 41$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45504, -162432, 276928, -276992, 169744, -58576, 5120, 6672, -3567, -104, 594, 132, -150, 40, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 + 40*x^13 - 150*x^12 + 132*x^11 + 594*x^10 - 104*x^9 - 3567*x^8 + 6672*x^7 + 5120*x^6 - 58576*x^5 + 169744*x^4 - 276992*x^3 + 276928*x^2 - 162432*x + 45504)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 + 40*x^13 - 150*x^12 + 132*x^11 + 594*x^10 - 104*x^9 - 3567*x^8 + 6672*x^7 + 5120*x^6 - 58576*x^5 + 169744*x^4 - 276992*x^3 + 276928*x^2 - 162432*x + 45504, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} + 40 x^{13} - 150 x^{12} + 132 x^{11} + 594 x^{10} - 104 x^{9} - 3567 x^{8} + 6672 x^{7} + 5120 x^{6} - 58576 x^{5} + 169744 x^{4} - 276992 x^{3} + 276928 x^{2} - 162432 x + 45504 \)

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:  \(199233812184432640000000000=2^{32}\cdot 5^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.03$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{1488} a^{12} - \frac{10}{93} a^{11} - \frac{29}{248} a^{10} + \frac{23}{93} a^{9} + \frac{85}{744} a^{8} + \frac{5}{124} a^{7} + \frac{25}{744} a^{6} + \frac{2}{31} a^{5} + \frac{227}{496} a^{4} - \frac{55}{372} a^{3} + \frac{17}{372} a^{2} - \frac{7}{93} a - \frac{7}{62}$, $\frac{1}{1488} a^{13} - \frac{53}{744} a^{11} + \frac{7}{186} a^{10} + \frac{137}{744} a^{9} - \frac{67}{372} a^{8} - \frac{11}{744} a^{7} - \frac{11}{186} a^{6} + \frac{139}{496} a^{5} - \frac{157}{372} a^{4} - \frac{67}{186} a^{3} + \frac{22}{93} a^{2} - \frac{29}{186} a - \frac{2}{31}$, $\frac{1}{830304} a^{14} + \frac{41}{415152} a^{13} - \frac{23}{207576} a^{12} + \frac{9371}{207576} a^{11} - \frac{18443}{138384} a^{10} - \frac{7265}{51894} a^{9} + \frac{12139}{415152} a^{8} + \frac{35263}{207576} a^{7} - \frac{46177}{276768} a^{6} - \frac{77405}{415152} a^{5} - \frac{59783}{138384} a^{4} + \frac{2705}{34596} a^{3} + \frac{11015}{25947} a^{2} - \frac{2417}{5766} a - \frac{1001}{5766}$, $\frac{1}{5130923104936647586368} a^{15} + \frac{14500750657237}{142525641803795766288} a^{14} + \frac{153242365531569307}{855153850822774597728} a^{13} - \frac{8876367007122335}{320682694058540474148} a^{12} - \frac{191417586606009840691}{2565461552468323793184} a^{11} - \frac{45914471114701230815}{1282730776234161896592} a^{10} - \frac{135695232236119258003}{2565461552468323793184} a^{9} + \frac{120312075505248422101}{641365388117080948296} a^{8} + \frac{18535315386025416203}{119323793138061571776} a^{7} + \frac{54088880752029472279}{641365388117080948296} a^{6} + \frac{188800600759589708347}{1282730776234161896592} a^{5} - \frac{8246358151170053543}{213788462705693649432} a^{4} + \frac{5878718472620350253}{20689206068292933816} a^{3} + \frac{22943065774455617801}{80170673514635118537} a^{2} - \frac{3145063316878289233}{8907852612737235393} a + \frac{2858735626162944548}{8907852612737235393}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $7$
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:  \( 507884.860663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), 4.0.65600.2, 4.0.65600.5, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.4303360000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$