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Properties

Label 16.0.43381651460...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}$
Root discriminant $61.64$
Ramified primes $2, 5, 11, 41$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![290639471, 51903780, -55097722, 66116880, 24944210, -28254140, -10234682, 3791040, 1697264, -205460, -133878, 4080, 5630, -20, -118, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 118*x^14 - 20*x^13 + 5630*x^12 + 4080*x^11 - 133878*x^10 - 205460*x^9 + 1697264*x^8 + 3791040*x^7 - 10234682*x^6 - 28254140*x^5 + 24944210*x^4 + 66116880*x^3 - 55097722*x^2 + 51903780*x + 290639471)
 
gp: K = bnfinit(x^16 - 118*x^14 - 20*x^13 + 5630*x^12 + 4080*x^11 - 133878*x^10 - 205460*x^9 + 1697264*x^8 + 3791040*x^7 - 10234682*x^6 - 28254140*x^5 + 24944210*x^4 + 66116880*x^3 - 55097722*x^2 + 51903780*x + 290639471, 1)
 

Normalized defining polynomial

\( x^{16} - 118 x^{14} - 20 x^{13} + 5630 x^{12} + 4080 x^{11} - 133878 x^{10} - 205460 x^{9} + 1697264 x^{8} + 3791040 x^{7} - 10234682 x^{6} - 28254140 x^{5} + 24944210 x^{4} + 66116880 x^{3} - 55097722 x^{2} + 51903780 x + 290639471 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43381651460325376000000000000=2^{32}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.64$
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:  $2, 5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{12} + \frac{1}{12} a^{11} + \frac{1}{24} a^{10} + \frac{1}{24} a^{9} - \frac{1}{8} a^{8} - \frac{1}{6} a^{7} + \frac{1}{8} a^{6} + \frac{5}{24} a^{5} - \frac{1}{24} a^{4} + \frac{5}{12} a^{3} + \frac{1}{8} a^{2} - \frac{5}{24} a - \frac{5}{24}$, $\frac{1}{6677883682764424916123527499966157053383551360168629064} a^{15} + \frac{67137409920223619121707200084183667326624070942844541}{6677883682764424916123527499966157053383551360168629064} a^{14} - \frac{43188312645437522533839239178296961884958871632022201}{6677883682764424916123527499966157053383551360168629064} a^{13} - \frac{47393662860641745011324216203728077728964173771489931}{1669470920691106229030881874991539263345887840042157266} a^{12} - \frac{279771694654872731270287616182112354652891663831080173}{6677883682764424916123527499966157053383551360168629064} a^{11} - \frac{73210745907872684367477924414143349744780336926279783}{2225961227588141638707842499988719017794517120056209688} a^{10} - \frac{249424563799486744233021184978472499829213944967915279}{6677883682764424916123527499966157053383551360168629064} a^{9} + \frac{7261941999701677410733929270341694906049408070675878}{834735460345553114515440937495769631672943920021078633} a^{8} + \frac{194406118428373977806390062952652447540560067513045295}{6677883682764424916123527499966157053383551360168629064} a^{7} - \frac{678568446041328851379422330773788448160144861088067813}{6677883682764424916123527499966157053383551360168629064} a^{6} + \frac{386050144473567432612150098270039172813734915557055283}{2225961227588141638707842499988719017794517120056209688} a^{5} - \frac{338509869646833955430433453957398974325036308705213477}{1669470920691106229030881874991539263345887840042157266} a^{4} - \frac{2188771232429563451715241931890251475596522745336874755}{6677883682764424916123527499966157053383551360168629064} a^{3} + \frac{1525488519918143332829847485555842126098932650653731533}{6677883682764424916123527499966157053383551360168629064} a^{2} + \frac{126940885058416608840366265638104608505621129246581365}{2225961227588141638707842499988719017794517120056209688} a + \frac{320873537624957113788510317458829968081148387177798332}{834735460345553114515440937495769631672943920021078633}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1675072028596792220699218755147534985801}{25964245525465392192146817745046470095773852973} a^{15} - \frac{1563204728371700657672498470029501255982}{8654748508488464064048939248348823365257950991} a^{14} - \frac{254232669863587796725395177230218658984765}{34618994033953856256195756993395293461031803964} a^{13} + \frac{1321655284831584408533583023985846191056387}{69237988067907712512391513986790586922063607928} a^{12} + \frac{5819341271011959133528133287402988936617331}{17309497016976928128097878496697646730515901982} a^{11} - \frac{67958003553192128903650411070009005307020003}{103856982101861568768587270980185880383095411892} a^{10} - \frac{827097866444403206666743607189181676396484603}{103856982101861568768587270980185880383095411892} a^{9} + \frac{1527809044746671509641010991862113370232524715}{207713964203723137537174541960371760766190823784} a^{8} + \frac{2916484990823502773646110021492482313642412236}{25964245525465392192146817745046470095773852973} a^{7} - \frac{757069637509178711138265625705369856386531559}{51928491050930784384293635490092940191547705946} a^{6} - \frac{86568432992337525427820138881605368883689026697}{103856982101861568768587270980185880383095411892} a^{5} - \frac{13296101000538602371957261048712589699713776239}{69237988067907712512391513986790586922063607928} a^{4} + \frac{137498546084379043890761869232080218279826160525}{51928491050930784384293635490092940191547705946} a^{3} - \frac{72469065117821188840708253750463598394166750215}{103856982101861568768587270980185880383095411892} a^{2} - \frac{181870017914511813022818572648530919231478895813}{103856982101861568768587270980185880383095411892} a + \frac{2244222312832116149089382532812605418700051290873}{207713964203723137537174541960371760766190823784} \) (order $10$)
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:  \( 25970554.6083 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41Data not computed