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Properties

Label 16.8.13886489905...7312.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 17^{8}\cdot 16673^{3}$
Root discriminant $102.07$
Ramified primes $2, 17, 16673$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1461

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63951713, 73520356, -49423048, -54882132, -16548400, -4576660, 982078, 1112084, 271449, 74908, -1766, -3300, -700, -192, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 18*x^14 - 192*x^13 - 700*x^12 - 3300*x^11 - 1766*x^10 + 74908*x^9 + 271449*x^8 + 1112084*x^7 + 982078*x^6 - 4576660*x^5 - 16548400*x^4 - 54882132*x^3 - 49423048*x^2 + 73520356*x + 63951713)
 
gp: K = bnfinit(x^16 + 18*x^14 - 192*x^13 - 700*x^12 - 3300*x^11 - 1766*x^10 + 74908*x^9 + 271449*x^8 + 1112084*x^7 + 982078*x^6 - 4576660*x^5 - 16548400*x^4 - 54882132*x^3 - 49423048*x^2 + 73520356*x + 63951713, 1)
 

Normalized defining polynomial

\( x^{16} + 18 x^{14} - 192 x^{13} - 700 x^{12} - 3300 x^{11} - 1766 x^{10} + 74908 x^{9} + 271449 x^{8} + 1112084 x^{7} + 982078 x^{6} - 4576660 x^{5} - 16548400 x^{4} - 54882132 x^{3} - 49423048 x^{2} + 73520356 x + 63951713 \)

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:  \(138864899053729330143760914317312=2^{32}\cdot 17^{8}\cdot 16673^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.07$
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:  $2, 17, 16673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{3}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} + \frac{1}{24} a^{12} - \frac{1}{16} a^{11} - \frac{11}{48} a^{10} + \frac{1}{6} a^{9} + \frac{3}{16} a^{8} + \frac{5}{16} a^{7} - \frac{5}{16} a^{6} + \frac{7}{24} a^{5} + \frac{11}{48} a^{4} - \frac{13}{48} a^{3} + \frac{1}{24} a^{2} - \frac{5}{16} a - \frac{23}{48}$, $\frac{1}{528738796738168423985919174782456514955886915655192144} a^{15} + \frac{278139515162665840061552054449877418658291946670673}{37767056909869173141851369627318322496849065403942296} a^{14} + \frac{28096894779033680698693892195222605346919271535291465}{528738796738168423985919174782456514955886915655192144} a^{13} + \frac{3819628129640126626752043714740538166863853880572373}{176246265579389474661973058260818838318628971885064048} a^{12} - \frac{1079220962432445788228633540348406221863522362923581}{5395293844267024734550195661045474642407009343420328} a^{11} - \frac{17953774802032413472109348712653100732345158037594613}{528738796738168423985919174782456514955886915655192144} a^{10} + \frac{3635963061811171287930217747891219549693454303694911}{176246265579389474661973058260818838318628971885064048} a^{9} + \frac{6442816616784107261907793558424544717834828988932669}{88123132789694737330986529130409419159314485942532024} a^{8} - \frac{160100878671098947091323709275078910015053518575647}{44061566394847368665493264565204709579657242971266012} a^{7} - \frac{191957806338240926535080759978188707244055596889661283}{528738796738168423985919174782456514955886915655192144} a^{6} + \frac{50520457206451279504515849147533505392197852011211301}{528738796738168423985919174782456514955886915655192144} a^{5} + \frac{26550891674640952915573619218148326754091797364517603}{66092349592271052998239896847807064369485864456899018} a^{4} - \frac{245512318661124228570171318351159881318994946425149479}{528738796738168423985919174782456514955886915655192144} a^{3} - \frac{12275886029225944671343664750085305727611745676161683}{176246265579389474661973058260818838318628971885064048} a^{2} + \frac{12086466339698505675513468636353751562445014620761543}{264369398369084211992959587391228257477943457827596072} a + \frac{9256530096028013307256791271628840829264613647605287}{25178037939912782094567579751545548331232710269294864}$

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:  \( 4334304947.13 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1461:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1461 are not computed
Character table for t16n1461 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), 4.4.9248.1 x2, 4.4.4352.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
16673Data not computed