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Properties

Label 16.8.34571164844...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 5^{15}\cdot 101^{3}$
Root discriminant $60.77$
Ramified primes $2, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1604

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9595, -40480, 28920, 18240, -23030, -14580, 24200, 53840, 41485, 16220, 2840, -440, -280, -60, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 60*x^13 - 280*x^12 - 440*x^11 + 2840*x^10 + 16220*x^9 + 41485*x^8 + 53840*x^7 + 24200*x^6 - 14580*x^5 - 23030*x^4 + 18240*x^3 + 28920*x^2 - 40480*x - 9595)
 
gp: K = bnfinit(x^16 - 20*x^14 - 60*x^13 - 280*x^12 - 440*x^11 + 2840*x^10 + 16220*x^9 + 41485*x^8 + 53840*x^7 + 24200*x^6 - 14580*x^5 - 23030*x^4 + 18240*x^3 + 28920*x^2 - 40480*x - 9595, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 60 x^{13} - 280 x^{12} - 440 x^{11} + 2840 x^{10} + 16220 x^{9} + 41485 x^{8} + 53840 x^{7} + 24200 x^{6} - 14580 x^{5} - 23030 x^{4} + 18240 x^{3} + 28920 x^{2} - 40480 x - 9595 \)

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:  \(34571164844032000000000000000=2^{40}\cdot 5^{15}\cdot 101^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.77$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{518913247415197779061876589850812974279} a^{15} + \frac{89650501352616622052302624241565469714}{518913247415197779061876589850812974279} a^{14} - \frac{211253114750667609493573269386339530079}{518913247415197779061876589850812974279} a^{13} + \frac{105902121567809793530829274076364360322}{518913247415197779061876589850812974279} a^{12} + \frac{135698971360910205795043726046085181190}{518913247415197779061876589850812974279} a^{11} + \frac{144945375058916428873449255381269371677}{518913247415197779061876589850812974279} a^{10} + \frac{180008289380264643712568524867503553963}{518913247415197779061876589850812974279} a^{9} + \frac{25113999015195882629907432302495592026}{518913247415197779061876589850812974279} a^{8} - \frac{181190251372778615145230224444147606917}{518913247415197779061876589850812974279} a^{7} + \frac{12563896140965525508042238290259099481}{518913247415197779061876589850812974279} a^{6} - \frac{258472360627026049184444494472335890855}{518913247415197779061876589850812974279} a^{5} + \frac{45210237031107674330744053285999107682}{518913247415197779061876589850812974279} a^{4} + \frac{144403069829372943457583381449446829123}{518913247415197779061876589850812974279} a^{3} + \frac{31312939210950154249275430199524776334}{518913247415197779061876589850812974279} a^{2} - \frac{197895518035676812613464407301988980879}{518913247415197779061876589850812974279} a + \frac{60340311090899877412563690534988003532}{518913247415197779061876589850812974279}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 378503900.238 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1604:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1604 are not computed
Character table for t16n1604 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
5Data not computed
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$