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Properties

Label 16.8.25893015832...5017.8
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 67^{6}$
Root discriminant $68.92$
Ramified primes $17, 67$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22064131, -5820344, -16468792, -1128470, -478367, 1017904, 1144659, 526530, 256954, 44662, 6995, 1642, -1169, -80, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 80*x^13 - 1169*x^12 + 1642*x^11 + 6995*x^10 + 44662*x^9 + 256954*x^8 + 526530*x^7 + 1144659*x^6 + 1017904*x^5 - 478367*x^4 - 1128470*x^3 - 16468792*x^2 - 5820344*x + 22064131)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 80*x^13 - 1169*x^12 + 1642*x^11 + 6995*x^10 + 44662*x^9 + 256954*x^8 + 526530*x^7 + 1144659*x^6 + 1017904*x^5 - 478367*x^4 - 1128470*x^3 - 16468792*x^2 - 5820344*x + 22064131, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 80 x^{13} - 1169 x^{12} + 1642 x^{11} + 6995 x^{10} + 44662 x^{9} + 256954 x^{8} + 526530 x^{7} + 1144659 x^{6} + 1017904 x^{5} - 478367 x^{4} - 1128470 x^{3} - 16468792 x^{2} - 5820344 x + 22064131 \)

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:  \(258930158322830089457985795017=17^{15}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.92$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{268} a^{13} - \frac{7}{67} a^{12} - \frac{10}{67} a^{11} - \frac{10}{67} a^{10} - \frac{13}{67} a^{9} - \frac{6}{67} a^{8} - \frac{51}{268} a^{7} - \frac{9}{134} a^{6} - \frac{7}{268} a^{5} + \frac{9}{134} a^{4} + \frac{91}{268} a^{3} - \frac{4}{67} a^{2} + \frac{25}{268} a - \frac{15}{67}$, $\frac{1}{536} a^{14} - \frac{1}{536} a^{13} - \frac{59}{536} a^{12} + \frac{43}{268} a^{11} - \frac{15}{134} a^{10} - \frac{11}{67} a^{9} - \frac{29}{536} a^{8} - \frac{55}{536} a^{7} - \frac{3}{67} a^{6} + \frac{97}{536} a^{5} + \frac{27}{134} a^{4} + \frac{163}{536} a^{3} + \frac{49}{134} a^{2} + \frac{79}{536} a + \frac{55}{536}$, $\frac{1}{462493910259713793504465736026767245093364136150008} a^{15} + \frac{44058016062871199425249959497954231842422529648}{57811738782464224188058217003345905636670517018751} a^{14} + \frac{141398094371936931735123728306648736302647077133}{115623477564928448376116434006691811273341034037502} a^{13} + \frac{17075539931313920081908881525313473365387714620731}{462493910259713793504465736026767245093364136150008} a^{12} + \frac{4050902241608928448469950733939602596039619034571}{17788227317681299750171759077952586349744774467308} a^{11} - \frac{3108072020293632462191344818949608552285341007465}{57811738782464224188058217003345905636670517018751} a^{10} - \frac{80503156912709736542888769498739495258552468188461}{462493910259713793504465736026767245093364136150008} a^{9} - \frac{12401143549178190646592001255750168782678455427350}{57811738782464224188058217003345905636670517018751} a^{8} - \frac{57486331692517422476824032896057696785133407587883}{462493910259713793504465736026767245093364136150008} a^{7} + \frac{115171152639491256811865903670595152666858506222373}{462493910259713793504465736026767245093364136150008} a^{6} + \frac{132065702752681452104005864270166016336745341885301}{462493910259713793504465736026767245093364136150008} a^{5} + \frac{180177873073778695693049664085504787982802010954159}{462493910259713793504465736026767245093364136150008} a^{4} + \frac{116087184341036495439762952767916285991332524706143}{462493910259713793504465736026767245093364136150008} a^{3} - \frac{152852595134237232984405410813219085052953884457293}{462493910259713793504465736026767245093364136150008} a^{2} - \frac{52047049340378457080871528168857319545117016202309}{231246955129856896752232868013383622546682068075004} a + \frac{67113865183369885446396328484765203393910919707771}{462493910259713793504465736026767245093364136150008}$

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:  $11$
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:  \( 127694604.371 \) (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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
67Data not computed