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Properties

Label 16.8.66283426716...3249.5
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 89^{8}$
Root discriminant $112.55$
Ramified primes $17, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1640683769, -292118775, -12814472, -88479221, -41026485, -182955, -5271253, 68341, 230388, 32681, 39923, -1176, -520, 142, -55, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 55*x^14 + 142*x^13 - 520*x^12 - 1176*x^11 + 39923*x^10 + 32681*x^9 + 230388*x^8 + 68341*x^7 - 5271253*x^6 - 182955*x^5 - 41026485*x^4 - 88479221*x^3 - 12814472*x^2 - 292118775*x + 1640683769)
 
gp: K = bnfinit(x^16 - 4*x^15 - 55*x^14 + 142*x^13 - 520*x^12 - 1176*x^11 + 39923*x^10 + 32681*x^9 + 230388*x^8 + 68341*x^7 - 5271253*x^6 - 182955*x^5 - 41026485*x^4 - 88479221*x^3 - 12814472*x^2 - 292118775*x + 1640683769, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 55 x^{14} + 142 x^{13} - 520 x^{12} - 1176 x^{11} + 39923 x^{10} + 32681 x^{9} + 230388 x^{8} + 68341 x^{7} - 5271253 x^{6} - 182955 x^{5} - 41026485 x^{4} - 88479221 x^{3} - 12814472 x^{2} - 292118775 x + 1640683769 \)

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:  \(662834267162184237456650608633249=17^{14}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.55$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{52} a^{13} - \frac{5}{52} a^{12} + \frac{1}{52} a^{11} + \frac{9}{52} a^{10} + \frac{3}{13} a^{9} + \frac{5}{52} a^{8} - \frac{1}{26} a^{7} + \frac{3}{52} a^{6} + \frac{9}{26} a^{5} + \frac{19}{52} a^{4} - \frac{19}{52} a^{3} - \frac{1}{26} a^{2} - \frac{1}{52} a + \frac{5}{13}$, $\frac{1}{208686396932} a^{14} + \frac{218269011}{208686396932} a^{13} - \frac{2723885679}{52171599233} a^{12} + \frac{300870397}{8026399882} a^{11} - \frac{31013908137}{208686396932} a^{10} + \frac{5715977030}{52171599233} a^{9} - \frac{7320764767}{104343198466} a^{8} - \frac{16171731143}{104343198466} a^{7} - \frac{23591772372}{52171599233} a^{6} - \frac{236197223}{4013199941} a^{5} - \frac{75850846885}{208686396932} a^{4} + \frac{7552601259}{16052799764} a^{3} + \frac{22285145859}{104343198466} a^{2} - \frac{18421535761}{104343198466} a - \frac{15766032319}{208686396932}$, $\frac{1}{20456561540634445734027386121644174247074368697959004} a^{15} - \frac{27496434131089416703926576618203934275095}{20456561540634445734027386121644174247074368697959004} a^{14} - \frac{80628526815641769924234683987373588230510321195373}{20456561540634445734027386121644174247074368697959004} a^{13} + \frac{170432722247861148656235045614529623412025629118616}{5114140385158611433506846530411043561768592174489751} a^{12} - \frac{323005984852329314464915019049107849223939348333513}{20456561540634445734027386121644174247074368697959004} a^{11} + \frac{765615102067381946351950804961062106709267893400197}{5114140385158611433506846530411043561768592174489751} a^{10} - \frac{972925619853062437109758763528843076540469316317507}{20456561540634445734027386121644174247074368697959004} a^{9} + \frac{1618662140536331242042492487356747267213268556807483}{20456561540634445734027386121644174247074368697959004} a^{8} + \frac{856288799048204109724473121884375680343806327790787}{20456561540634445734027386121644174247074368697959004} a^{7} + \frac{9082164768475487935936196904234313260318800976465467}{20456561540634445734027386121644174247074368697959004} a^{6} + \frac{377530861967055598575937899735159798305201158631067}{5114140385158611433506846530411043561768592174489751} a^{5} - \frac{4182828922533966748948786422725802883554730544940893}{10228280770317222867013693060822087123537184348979502} a^{4} + \frac{1080333127030506687585274259291634922097025325669787}{10228280770317222867013693060822087123537184348979502} a^{3} + \frac{7461924913430976616998055628953396379266941337348609}{20456561540634445734027386121644174247074368697959004} a^{2} - \frac{2837072605068695400042499913128178864196112910969511}{10228280770317222867013693060822087123537184348979502} a + \frac{9276104448429115556835846876600551083485742731680087}{20456561540634445734027386121644174247074368697959004}$

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:  \( 16786016205.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.25745567912986193.1, 8.4.289276043966137.1, 8.4.2148243641.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$