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Properties

Label 16.8.11420589882...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{38}\cdot 5^{8}\cdot 89^{4}\cdot 130201^{2}$
Root discriminant $155.28$
Ramified primes $2, 5, 89, 130201$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67809201604, 0, -10571279592, 0, -2081661750, 0, 88066646, 0, 13923457, 0, 60096, 0, -7343, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 7343*x^12 + 60096*x^10 + 13923457*x^8 + 88066646*x^6 - 2081661750*x^4 - 10571279592*x^2 + 67809201604)
 
gp: K = bnfinit(x^16 - 20*x^14 - 7343*x^12 + 60096*x^10 + 13923457*x^8 + 88066646*x^6 - 2081661750*x^4 - 10571279592*x^2 + 67809201604, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 7343 x^{12} + 60096 x^{10} + 13923457 x^{8} + 88066646 x^{6} - 2081661750 x^{4} - 10571279592 x^{2} + 67809201604 \)

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:  \(114205898821156016581142118400000000=2^{38}\cdot 5^{8}\cdot 89^{4}\cdot 130201^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.28$
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:  $2, 5, 89, 130201$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{8} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{110171871538833990694210208113316217342} a^{14} + \frac{1787370880006017450019492437223562491}{36723957179611330231403402704438739114} a^{12} - \frac{180915810447297495168024726003122165}{36723957179611330231403402704438739114} a^{10} + \frac{113296122392222360807875265937325087}{110171871538833990694210208113316217342} a^{8} - \frac{2076865309165324003886532487128477017}{36723957179611330231403402704438739114} a^{6} - \frac{3626099353927685295124462532050904329}{110171871538833990694210208113316217342} a^{4} - \frac{25727944524140383465450702758261775385}{55085935769416995347105104056658108671} a^{2} - \frac{100734300970341370252077592537246}{423083814789571472931122679984471}$, $\frac{1}{110171871538833990694210208113316217342} a^{15} + \frac{1787370880006017450019492437223562491}{36723957179611330231403402704438739114} a^{13} - \frac{180915810447297495168024726003122165}{36723957179611330231403402704438739114} a^{11} + \frac{113296122392222360807875265937325087}{110171871538833990694210208113316217342} a^{9} - \frac{2076865309165324003886532487128477017}{36723957179611330231403402704438739114} a^{7} - \frac{3626099353927685295124462532050904329}{110171871538833990694210208113316217342} a^{5} - \frac{25727944524140383465450702758261775385}{55085935769416995347105104056658108671} a^{3} - \frac{100734300970341370252077592537246}{423083814789571472931122679984471} a$

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:  \( 559989557984 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.20.42$x^{8} + 16 x^{7} + 8 x^{6} + 240$$4$$2$$20$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.18.48$x^{8} + 6 x^{6} + 2 x^{4} + 12$$4$$2$$18$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
5Data not computed
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
130201Data not computed