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Properties

Label 16.8.31236842166...5728.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{15}\cdot 6529^{2}$
Root discriminant $60.38$
Ramified primes $2, 17, 6529$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7366864, -14923984, -6494240, -3724040, -3147512, -85776, 258154, -3888, 65279, 14151, -3636, 768, -242, -91, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 - 91*x^13 - 242*x^12 + 768*x^11 - 3636*x^10 + 14151*x^9 + 65279*x^8 - 3888*x^7 + 258154*x^6 - 85776*x^5 - 3147512*x^4 - 3724040*x^3 - 6494240*x^2 - 14923984*x - 7366864)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 - 91*x^13 - 242*x^12 + 768*x^11 - 3636*x^10 + 14151*x^9 + 65279*x^8 - 3888*x^7 + 258154*x^6 - 85776*x^5 - 3147512*x^4 - 3724040*x^3 - 6494240*x^2 - 14923984*x - 7366864, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} - 91 x^{13} - 242 x^{12} + 768 x^{11} - 3636 x^{10} + 14151 x^{9} + 65279 x^{8} - 3888 x^{7} + 258154 x^{6} - 85776 x^{5} - 3147512 x^{4} - 3724040 x^{3} - 6494240 x^{2} - 14923984 x - 7366864 \)

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:  \(31236842166910780816202985728=2^{8}\cdot 17^{15}\cdot 6529^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.38$
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:  $2, 17, 6529$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{346492396975328743011315206095421065474605141563048} a^{15} + \frac{11980733794485912091401653953836058462901318037601}{346492396975328743011315206095421065474605141563048} a^{14} + \frac{595758177345500125023508182274186597009184589915}{173246198487664371505657603047710532737302570781524} a^{13} + \frac{29478624347714513249867116520552358014491091217001}{346492396975328743011315206095421065474605141563048} a^{12} - \frac{4758458867923440894137977742696989123694091162993}{43311549621916092876414400761927633184325642695381} a^{11} - \frac{7822523928321951716847481003134630648172640399709}{86623099243832185752828801523855266368651285390762} a^{10} + \frac{344359439129218048094739691600508679608984521605}{86623099243832185752828801523855266368651285390762} a^{9} + \frac{24058107219643919724059449811028998448829073509563}{346492396975328743011315206095421065474605141563048} a^{8} - \frac{139770407860406535085528941222807458921275756341135}{346492396975328743011315206095421065474605141563048} a^{7} + \frac{68518273674331330468221617523919180079596845537303}{173246198487664371505657603047710532737302570781524} a^{6} - \frac{71925364548228295477179593868941973699046051860117}{173246198487664371505657603047710532737302570781524} a^{5} - \frac{18575113697623281812010546625880728677619839048012}{43311549621916092876414400761927633184325642695381} a^{4} - \frac{15314961940700687973043315942624435378942493715789}{86623099243832185752828801523855266368651285390762} a^{3} + \frac{11951188408216714604021418476886097115859709716413}{43311549621916092876414400761927633184325642695381} a^{2} + \frac{16254234689271959234554976478850150569225105397042}{43311549621916092876414400761927633184325642695381} a - \frac{14431471745461256362327901086359993596002694254441}{43311549621916092876414400761927633184325642695381}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 74121395.6108 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
6529Data not computed