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Properties

Label 16.8.25893015832...5017.9
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 67^{6}$
Root discriminant $68.92$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27895267, 34884977, -28965335, -33994711, -22529999, -7489970, -255115, 272440, 235026, 17212, 5657, 461, -817, 9, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 13*x^14 + 9*x^13 - 817*x^12 + 461*x^11 + 5657*x^10 + 17212*x^9 + 235026*x^8 + 272440*x^7 - 255115*x^6 - 7489970*x^5 - 22529999*x^4 - 33994711*x^3 - 28965335*x^2 + 34884977*x + 27895267)
 
gp: K = bnfinit(x^16 - 2*x^15 - 13*x^14 + 9*x^13 - 817*x^12 + 461*x^11 + 5657*x^10 + 17212*x^9 + 235026*x^8 + 272440*x^7 - 255115*x^6 - 7489970*x^5 - 22529999*x^4 - 33994711*x^3 - 28965335*x^2 + 34884977*x + 27895267, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 13 x^{14} + 9 x^{13} - 817 x^{12} + 461 x^{11} + 5657 x^{10} + 17212 x^{9} + 235026 x^{8} + 272440 x^{7} - 255115 x^{6} - 7489970 x^{5} - 22529999 x^{4} - 33994711 x^{3} - 28965335 x^{2} + 34884977 x + 27895267 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{72628} a^{14} - \frac{3475}{18157} a^{13} + \frac{363}{36314} a^{12} - \frac{2819}{72628} a^{11} + \frac{6113}{72628} a^{10} + \frac{5993}{36314} a^{9} + \frac{2521}{36314} a^{8} + \frac{6118}{18157} a^{7} - \frac{2717}{36314} a^{6} - \frac{8246}{18157} a^{5} - \frac{2567}{72628} a^{4} - \frac{14095}{36314} a^{3} - \frac{2943}{36314} a^{2} + \frac{8397}{72628} a - \frac{6153}{72628}$, $\frac{1}{370139751278135222358119429002444713672798086394812584} a^{15} + \frac{1621695981435394086480886102873441086605335418951}{370139751278135222358119429002444713672798086394812584} a^{14} - \frac{2589184619580017932237326703379743750872993671614707}{185069875639067611179059714501222356836399043197406292} a^{13} - \frac{21978962651530020996964929388743305801217515056893021}{370139751278135222358119429002444713672798086394812584} a^{12} - \frac{35057187056655013057492648337987365425251110669049357}{185069875639067611179059714501222356836399043197406292} a^{11} - \frac{18171023125320934891128580409756564265056706152566609}{370139751278135222358119429002444713672798086394812584} a^{10} + \frac{2519779912085328872182322252643196358238167910877675}{46267468909766902794764928625305589209099760799351573} a^{9} + \frac{15055263328637923542141647417872649413960686681664510}{46267468909766902794764928625305589209099760799351573} a^{8} - \frac{60766779464772241949683970173204451227714715661668175}{185069875639067611179059714501222356836399043197406292} a^{7} + \frac{46371975029724405365152179869615520940996567438723505}{185069875639067611179059714501222356836399043197406292} a^{6} - \frac{107582840032903075708343628010493243936683684550412213}{370139751278135222358119429002444713672798086394812584} a^{5} - \frac{76058338618113317195085778143031975676291726199551519}{370139751278135222358119429002444713672798086394812584} a^{4} - \frac{7385788300519111587724978506221796519079896024432669}{185069875639067611179059714501222356836399043197406292} a^{3} - \frac{140389895831033474719934214566433713983374523609593313}{370139751278135222358119429002444713672798086394812584} a^{2} - \frac{13193585606037625616600189010813596606288654025904111}{92534937819533805589529857250611178418199521598703146} a - \frac{172716309516669563925606003417587031987450365386451987}{370139751278135222358119429002444713672798086394812584}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 66056272.4874 \) (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