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Properties

Label 16.8.28020861582...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}$
Root discriminant $79.98$
Ramified primes $2, 5, 101, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-826564, 4399080, -5902568, 716008, 675772, -127328, 139904, -286176, -18510, 8816, 4892, 2724, -648, -48, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 - 48*x^13 - 648*x^12 + 2724*x^11 + 4892*x^10 + 8816*x^9 - 18510*x^8 - 286176*x^7 + 139904*x^6 - 127328*x^5 + 675772*x^4 + 716008*x^3 - 5902568*x^2 + 4399080*x - 826564)
 
gp: K = bnfinit(x^16 - 6*x^15 + 12*x^14 - 48*x^13 - 648*x^12 + 2724*x^11 + 4892*x^10 + 8816*x^9 - 18510*x^8 - 286176*x^7 + 139904*x^6 - 127328*x^5 + 675772*x^4 + 716008*x^3 - 5902568*x^2 + 4399080*x - 826564, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 12 x^{14} - 48 x^{13} - 648 x^{12} + 2724 x^{11} + 4892 x^{10} + 8816 x^{9} - 18510 x^{8} - 286176 x^{7} + 139904 x^{6} - 127328 x^{5} + 675772 x^{4} + 716008 x^{3} - 5902568 x^{2} + 4399080 x - 826564 \)

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:  \(2802086158223343616000000000000=2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.98$
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:  $2, 5, 101, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{684961323147928488521558459077999354715495683703882} a^{15} - \frac{9257614317668012439436874146783555681610770329927}{342480661573964244260779229538999677357747841851941} a^{14} + \frac{123183846343262360162572361435442109629313617916411}{684961323147928488521558459077999354715495683703882} a^{13} + \frac{82771458351272139561746473244181165660920928796944}{342480661573964244260779229538999677357747841851941} a^{12} + \frac{3055883853357863888740886592000362618194755954539}{684961323147928488521558459077999354715495683703882} a^{11} + \frac{64768824353684723566871749652896607592310678025889}{342480661573964244260779229538999677357747841851941} a^{10} - \frac{20972023595727154485468990548336310687312801878287}{342480661573964244260779229538999677357747841851941} a^{9} - \frac{71343398682526376934391303463235171225551175703528}{342480661573964244260779229538999677357747841851941} a^{8} - \frac{141770108614286532958527265995650229780519979823756}{342480661573964244260779229538999677357747841851941} a^{7} - \frac{112376750683168778764327262479170229835959363003672}{342480661573964244260779229538999677357747841851941} a^{6} - \frac{156986972091535114604141159197851100354567784852618}{342480661573964244260779229538999677357747841851941} a^{5} + \frac{44412792161632982067038116206810769333128916902173}{342480661573964244260779229538999677357747841851941} a^{4} - \frac{59567174102821396027856667210636019467075905217807}{342480661573964244260779229538999677357747841851941} a^{3} + \frac{163861119396386817100158012027303664366674366850851}{342480661573964244260779229538999677357747841851941} a^{2} - \frac{85451280018142966782310980645414453672851321478563}{342480661573964244260779229538999677357747841851941} a + \frac{65936076070582219968223583063893220518766534707390}{342480661573964244260779229538999677357747841851941}$

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:  \( 2299020529.12 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$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}$
101Data not computed
641Data not computed