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Properties

Label 16.8.54495238003...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{8}\cdot 11^{4}\cdot 13^{4}\cdot 19^{4}$
Root discriminant $22.83$
Ramified primes $2, 5, 11, 13, 19$
Class number $1$
Class group Trivial
Galois group 16T1553

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -64, 412, -810, 365, 1425, -3696, 4859, -4267, 2651, -1132, 266, 32, -60, 29, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 29*x^14 - 60*x^13 + 32*x^12 + 266*x^11 - 1132*x^10 + 2651*x^9 - 4267*x^8 + 4859*x^7 - 3696*x^6 + 1425*x^5 + 365*x^4 - 810*x^3 + 412*x^2 - 64*x - 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 29*x^14 - 60*x^13 + 32*x^12 + 266*x^11 - 1132*x^10 + 2651*x^9 - 4267*x^8 + 4859*x^7 - 3696*x^6 + 1425*x^5 + 365*x^4 - 810*x^3 + 412*x^2 - 64*x - 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 29 x^{14} - 60 x^{13} + 32 x^{12} + 266 x^{11} - 1132 x^{10} + 2651 x^{9} - 4267 x^{8} + 4859 x^{7} - 3696 x^{6} + 1425 x^{5} + 365 x^{4} - 810 x^{3} + 412 x^{2} - 64 x - 4 \)

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:  \(5449523800392100000000=2^{8}\cdot 5^{8}\cdot 11^{4}\cdot 13^{4}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.83$
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:  $2, 5, 11, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{104} a^{14} - \frac{11}{104} a^{13} - \frac{15}{104} a^{12} + \frac{43}{104} a^{10} + \frac{33}{104} a^{9} + \frac{17}{52} a^{8} + \frac{1}{13} a^{7} - \frac{45}{104} a^{6} + \frac{5}{52} a^{5} + \frac{51}{104} a^{4} - \frac{9}{52} a^{3} + \frac{7}{26} a^{2} - \frac{7}{26} a + \frac{1}{26}$, $\frac{1}{6785667594472} a^{15} + \frac{24830982129}{6785667594472} a^{14} + \frac{124836337201}{6785667594472} a^{13} + \frac{274511575473}{1696416898618} a^{12} + \frac{211089747275}{6785667594472} a^{11} + \frac{558477135001}{6785667594472} a^{10} + \frac{614586135145}{3392833797236} a^{9} - \frac{409597845250}{848208449309} a^{8} - \frac{732748889777}{6785667594472} a^{7} + \frac{83138923331}{3392833797236} a^{6} - \frac{494836597473}{6785667594472} a^{5} + \frac{670394157559}{3392833797236} a^{4} + \frac{6243051912}{65246803793} a^{3} - \frac{242219643438}{848208449309} a^{2} - \frac{433270366055}{1696416898618} a + \frac{272361655405}{848208449309}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 129327.569503 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1553:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 106 conjugacy class representatives for t16n1553 are not computed
Character table for t16n1553 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5225.1, 8.6.436810000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$