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Properties

Label 16.8.42832921613...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{8}\cdot 17^{2}\cdot 19^{10}\cdot 331^{8}$
Root discriminant $1460.45$
Ramified primes $2, 5, 17, 19, 331$
Class number $48$ (GRH)
Class group $[2, 2, 2, 6]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![592949281024, 0, 9042674097920, 0, 2863944241360, 0, -190660506576, 0, 972187419, 0, 28031400, 0, -110144, 0, -808, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 808*x^14 - 110144*x^12 + 28031400*x^10 + 972187419*x^8 - 190660506576*x^6 + 2863944241360*x^4 + 9042674097920*x^2 + 592949281024)
 
gp: K = bnfinit(x^16 - 808*x^14 - 110144*x^12 + 28031400*x^10 + 972187419*x^8 - 190660506576*x^6 + 2863944241360*x^4 + 9042674097920*x^2 + 592949281024, 1)
 

Normalized defining polynomial

\( x^{16} - 808 x^{14} - 110144 x^{12} + 28031400 x^{10} + 972187419 x^{8} - 190660506576 x^{6} + 2863944241360 x^{4} + 9042674097920 x^{2} + 592949281024 \)

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:  \(428329216131304888705048916200034767627878400000000=2^{32}\cdot 5^{8}\cdot 17^{2}\cdot 19^{10}\cdot 331^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1460.45$
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:  $2, 5, 17, 19, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a$, $\frac{1}{36} a^{10} - \frac{1}{18} a^{8} + \frac{1}{18} a^{6} + \frac{7}{18} a^{4} + \frac{1}{36} a^{2} + \frac{4}{9}$, $\frac{1}{72} a^{11} + \frac{1}{18} a^{9} - \frac{1}{18} a^{7} + \frac{1}{9} a^{5} + \frac{31}{72} a^{3} + \frac{1}{18} a$, $\frac{1}{232560} a^{12} + \frac{7}{58140} a^{10} - \frac{295}{5814} a^{8} + \frac{781}{29070} a^{6} - \frac{173}{232560} a^{4} + \frac{349}{3060} a^{2} - \frac{2}{5}$, $\frac{1}{465120} a^{13} + \frac{7}{116280} a^{11} + \frac{337}{5814} a^{9} + \frac{781}{58140} a^{7} + \frac{116107}{465120} a^{5} + \frac{349}{6120} a^{3} - \frac{1}{30} a$, $\frac{1}{12242639260746035936540274061962694360434240} a^{14} + \frac{150332823227132947531415609656990379}{139120900690295862915230387067757890459480} a^{12} - \frac{4618495134328026122717037275913484330909}{382582476898313623016883564436334198763570} a^{10} - \frac{13409751915925085862420733793524924113359}{1530329907593254492067534257745336795054280} a^{8} + \frac{558331428342904953378449869099851837917339}{12242639260746035936540274061962694360434240} a^{6} + \frac{11175851720752248256515403316655854903803}{22504851582253742530404915555078482280210} a^{4} - \frac{8846020220874038956476665623941059097787}{40271839673506697159671954151193073554060} a^{2} - \frac{14968476943127291121727303725267327058}{592232936375098487642234619870486375795}$, $\frac{1}{3648306499702318709089001670464882919409403520} a^{15} + \frac{28864657011110695984730669906407844963}{41458028405708167148738655346191851356925040} a^{13} + \frac{909132043802249449224849792318893248879313}{228019156231394919318062604404055182463087720} a^{11} + \frac{2522241226508915671650415081553009864060389}{50670923606976648737347245423123373880686160} a^{9} - \frac{10877818610293282103320062605556683630315233}{729661299940463741817800334092976583881880704} a^{7} - \frac{4685586678150816529268957098741058823120167}{19001596352616243276505217033671265205257310} a^{5} + \frac{11697320487564798268653955231043874323049}{66672267903916643075456901872530755106166} a^{3} - \frac{53565986657171477207038453264903246507277}{176485415039779349317385916721404939986910} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (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:  \( 1743645359930000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{31445}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6289}) \), 4.4.7600.1, 4.4.832663600.1, \(\Q(\sqrt{5}, \sqrt{6289})\), 8.8.250291650146150560000.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.51$x^{8} + 3$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
2.8.16.49$x^{8} + 14 x^{4} + 12 x^{2} + 8 x + 4$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
331Data not computed