↑

Properties

Label 16.8.63493735177...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{12}\cdot 5^{8}\cdot 41^{8}\cdot 89^{6}$
Root discriminant $129.62$
Ramified primes $2, 5, 41, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $D_4.C_2^3.C_2$ (as 16T264)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![201601, -13014714, -192536149, 279583018, 124706865, -3507946, -4831565, 571692, 417084, 2618, -12663, 3946, 1873, -118, -87, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 87*x^14 - 118*x^13 + 1873*x^12 + 3946*x^11 - 12663*x^10 + 2618*x^9 + 417084*x^8 + 571692*x^7 - 4831565*x^6 - 3507946*x^5 + 124706865*x^4 + 279583018*x^3 - 192536149*x^2 - 13014714*x + 201601)
 
gp: K = bnfinit(x^16 - 87*x^14 - 118*x^13 + 1873*x^12 + 3946*x^11 - 12663*x^10 + 2618*x^9 + 417084*x^8 + 571692*x^7 - 4831565*x^6 - 3507946*x^5 + 124706865*x^4 + 279583018*x^3 - 192536149*x^2 - 13014714*x + 201601, 1)
 

Normalized defining polynomial

\( x^{16} - 87 x^{14} - 118 x^{13} + 1873 x^{12} + 3946 x^{11} - 12663 x^{10} + 2618 x^{9} + 417084 x^{8} + 571692 x^{7} - 4831565 x^{6} - 3507946 x^{5} + 124706865 x^{4} + 279583018 x^{3} - 192536149 x^{2} - 13014714 x + 201601 \)

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:  \(6349373517752981266040449600000000=2^{12}\cdot 5^{8}\cdot 41^{8}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.62$
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:  $2, 5, 41, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{3}{16} a^{2} + \frac{3}{8} a - \frac{1}{16}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{3}{32} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{5}{32} a^{3} + \frac{1}{32} a^{2} + \frac{9}{32} a - \frac{7}{32}$, $\frac{1}{160} a^{14} + \frac{1}{160} a^{13} - \frac{1}{32} a^{12} - \frac{17}{160} a^{11} + \frac{13}{80} a^{10} - \frac{1}{8} a^{9} + \frac{7}{40} a^{8} + \frac{3}{80} a^{7} - \frac{11}{80} a^{6} + \frac{7}{20} a^{5} + \frac{59}{160} a^{4} - \frac{1}{160} a^{3} + \frac{23}{160} a^{2} + \frac{7}{32} a - \frac{13}{80}$, $\frac{1}{735896934131366645720381615350185251630000709562747726091840} a^{15} + \frac{448401115844276132992628680284002018909089513279067225}{327793734579673338850949494588055791371937955261802996032} a^{14} - \frac{2179352358787188835347530316196045698996574596007089490149}{183974233532841661430095403837546312907500177390686931522960} a^{13} + \frac{9117221886111799965110111858572030173697009033041536884289}{367948467065683322860190807675092625815000354781373863045920} a^{12} + \frac{4623078760913814707990685256666196963486640888085296134333}{23738610778431182120012310172586621020322603534282184712640} a^{11} - \frac{1818459264211022447899391664904442196573468174545309220611}{12063884166087977798694780579511233633278700156766356165440} a^{10} + \frac{22045804163107223585175251360178311939061609867195814626141}{91987116766420830715047701918773156453750088695343465761480} a^{9} + \frac{71122954621758614602893255250217561867847851044076296603669}{367948467065683322860190807675092625815000354781373863045920} a^{8} - \frac{27067260780039620890035081762769228627341669573491434926749}{367948467065683322860190807675092625815000354781373863045920} a^{7} - \frac{27807467224093461343548447341948909539324583700287249101221}{367948467065683322860190807675092625815000354781373863045920} a^{6} + \frac{19832457457246920976308447171349105467837726830595798779513}{735896934131366645720381615350185251630000709562747726091840} a^{5} + \frac{10579124153165666310118685026835899869564483070248597098907}{147179386826273329144076323070037050326000141912549545218368} a^{4} + \frac{138167497398773307145985103768683344518757357733789454816637}{367948467065683322860190807675092625815000354781373863045920} a^{3} + \frac{4820063397198113397841891566412159621166599831004465672967}{45993558383210415357523850959386578226875044347671732880740} a^{2} - \frac{44674071203294634472468814208137036175729856933120608348321}{735896934131366645720381615350185251630000709562747726091840} a - \frac{77068354074156913615328803391928178181991984848496936431}{1638968672898366694254747472940278956859689776309014980160}$

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:  \( 23039056748.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4.C_2^3.C_2$ (as 16T264):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $D_4.C_2^3.C_2$
Character table for $D_4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{41}) \), 4.4.2225.1, 4.4.3740225.1, \(\Q(\sqrt{5}, \sqrt{41})\), 8.8.13989283050625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$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}$
5.4.2.1$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}$
41Data not computed
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$