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Properties

Label 16.0.38693032794...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{2}\cdot 29^{4}\cdot 41^{4}$
Root discriminant $52.99$
Ramified primes $2, 5, 11, 29, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![341917081, 0, -62924873, 0, 12864693, 0, -1399986, 0, 162415, 0, -11316, 0, 788, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 28*x^14 + 788*x^12 - 11316*x^10 + 162415*x^8 - 1399986*x^6 + 12864693*x^4 - 62924873*x^2 + 341917081)
 
gp: K = bnfinit(x^16 - 28*x^14 + 788*x^12 - 11316*x^10 + 162415*x^8 - 1399986*x^6 + 12864693*x^4 - 62924873*x^2 + 341917081, 1)
 

Normalized defining polynomial

\( x^{16} - 28 x^{14} + 788 x^{12} - 11316 x^{10} + 162415 x^{8} - 1399986 x^{6} + 12864693 x^{4} - 62924873 x^{2} + 341917081 \)

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:  \(3869303279468176000000000000=2^{16}\cdot 5^{12}\cdot 11^{2}\cdot 29^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.99$
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:  $2, 5, 11, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{205} a^{10} + \frac{13}{205} a^{8} - \frac{32}{205} a^{6} - \frac{2}{5} a^{4} - \frac{27}{205} a^{2} + \frac{2}{5}$, $\frac{1}{205} a^{11} + \frac{13}{205} a^{9} - \frac{32}{205} a^{7} - \frac{2}{5} a^{5} - \frac{27}{205} a^{3} + \frac{2}{5} a$, $\frac{1}{8405} a^{12} + \frac{13}{8405} a^{10} - \frac{72}{1681} a^{8} + \frac{4}{41} a^{6} + \frac{544}{1681} a^{4} - \frac{61}{205} a^{2} + \frac{2}{5}$, $\frac{1}{8405} a^{13} + \frac{13}{8405} a^{11} - \frac{72}{1681} a^{9} + \frac{4}{41} a^{7} + \frac{544}{1681} a^{5} - \frac{61}{205} a^{3} + \frac{2}{5} a$, $\frac{1}{8464659248770964305} a^{14} - \frac{419761897660067}{8464659248770964305} a^{12} - \frac{15024804722212102}{8464659248770964305} a^{10} + \frac{17239469220705291}{206455103628560105} a^{8} + \frac{335075140151379326}{769514477160996755} a^{6} - \frac{17812183607827363}{41291020725712021} a^{4} - \frac{2190661927670692}{5035490332403905} a^{2} - \frac{4169758503772}{11165167034155}$, $\frac{1}{8464659248770964305} a^{15} - \frac{419761897660067}{8464659248770964305} a^{13} - \frac{15024804722212102}{8464659248770964305} a^{11} + \frac{17239469220705291}{206455103628560105} a^{9} + \frac{335075140151379326}{769514477160996755} a^{7} - \frac{17812183607827363}{41291020725712021} a^{5} - \frac{2190661927670692}{5035490332403905} a^{3} - \frac{4169758503772}{11165167034155} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{2369898331844}{8464659248770964305} a^{14} - \frac{57412166335672}{8464659248770964305} a^{12} + \frac{1247525506214277}{8464659248770964305} a^{10} - \frac{72050231252012}{41291020725712021} a^{8} + \frac{2954622257190723}{153902895432199351} a^{6} - \frac{27952530724340979}{206455103628560105} a^{4} + \frac{4417096121600407}{5035490332403905} a^{2} - \frac{18350365920902}{11165167034155} \) (order $10$)
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:  \( 19911328.1321 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.3625.1, 4.4.725.1, 8.0.13140625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed