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Properties

Label 16.16.1623938554...3136.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{26}\cdot 193^{8}\cdot 257^{10}$
Root discriminant $1374.55$
Ramified primes $2, 193, 257$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28613530403758336, 0, -3937312460664784, 0, 197363433219140, 0, -4480851471638, 0, 49105808800, 0, -286021943, 0, 911285, 0, -1501, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1501*x^14 + 911285*x^12 - 286021943*x^10 + 49105808800*x^8 - 4480851471638*x^6 + 197363433219140*x^4 - 3937312460664784*x^2 + 28613530403758336)
 
gp: K = bnfinit(x^16 - 1501*x^14 + 911285*x^12 - 286021943*x^10 + 49105808800*x^8 - 4480851471638*x^6 + 197363433219140*x^4 - 3937312460664784*x^2 + 28613530403758336, 1)
 

Normalized defining polynomial

\( x^{16} - 1501 x^{14} + 911285 x^{12} - 286021943 x^{10} + 49105808800 x^{8} - 4480851471638 x^{6} + 197363433219140 x^{4} - 3937312460664784 x^{2} + 28613530403758336 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(162393855437479514050983820757859401106269862363136=2^{26}\cdot 193^{8}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1374.55$
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:  $2, 193, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1028} a^{12} + \frac{41}{1028} a^{10} - \frac{37}{1028} a^{8} + \frac{39}{1028} a^{6} - \frac{1}{514} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2056} a^{13} + \frac{41}{2056} a^{11} - \frac{1}{4} a^{10} + \frac{477}{2056} a^{9} + \frac{39}{2056} a^{7} - \frac{1}{4} a^{6} - \frac{129}{514} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{146376616003587953345285758302231242357866064} a^{14} - \frac{2514677552300413884080672300482163965439}{13306965091235268485935068936566476577987824} a^{12} - \frac{16553310607123634938406116481805538407475747}{146376616003587953345285758302231242357866064} a^{10} + \frac{1542811880110022611285738468532959024306471}{20910945143369707620755108328890177479695152} a^{8} + \frac{2074365488489736312593882243184488780042055}{18297077000448494168160719787778905294733258} a^{6} - \frac{3911727798767951075295631133705992534385189}{10455472571684853810377554164445088739847576} a^{4} + \frac{3972659142409621582666252219291052688017}{142389704283645869012923889399057628752788} a^{2} - \frac{1618312637428502606288417634014643743357}{35597426070911467253230972349764407188197}$, $\frac{1}{48171958820316781094120161914231092935004290198144} a^{15} - \frac{1}{292753232007175906690571516604462484715732128} a^{14} + \frac{378003316639813760214788957348725051912299577}{4379268983665161917647287446748281175909480927104} a^{13} + \frac{2514677552300413884080672300482163965439}{26613930182470536971870137873132953155975648} a^{12} + \frac{594370756559915682596540690074262388198368167813}{48171958820316781094120161914231092935004290198144} a^{11} - \frac{56634997394670341734236762669310082771457285}{292753232007175906690571516604462484715732128} a^{10} + \frac{231568413625785245399142699015610878847608044255}{6881708402902397299160023130604441847857755742592} a^{9} - \frac{1542811880110022611285738468532959024306471}{41821890286739415241510216657780354959390304} a^{8} - \frac{887262120029055722602848069938343849708677495099}{3010747426269798818382510119639443308437768137384} a^{7} - \frac{2805725997178495849168560534268485356852171}{9148538500224247084080359893889452647366629} a^{6} - \frac{577698184958014676963948404351592149916423892253}{3440854201451198649580011565302220923928877871296} a^{5} + \frac{3911727798767951075295631133705992534385189}{20910945143369707620755108328890177479695152} a^{4} - \frac{12676381840896714725122979511388285166293016471}{46859882120930720908677200305672269392027519648} a^{3} - \frac{3972659142409621582666252219291052688017}{284779408567291738025847778798115257505576} a^{2} - \frac{3601464408758892393446490222987998618688010241}{11714970530232680227169300076418067348006879912} a + \frac{1618312637428502606288417634014643743357}{71194852141822934506461944699528814376394}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  $15$
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:  \( 855808015520000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), \(\Q(\sqrt{193}) \), \(\Q(\sqrt{49601}) \), 4.4.19682073608.1, 4.4.528392.1, \(\Q(\sqrt{193}, \sqrt{257})\), 8.8.387384021510730137664.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.11.15$x^{4} + 30$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
193Data not computed
257Data not computed