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Properties

Label 16.0.91778367105...8144.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $36.32$
Ramified primes $2, 17, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![162641, 161516, -138942, -72900, 140344, 15732, -40238, 5144, 8925, -3900, -1202, 584, 132, -40, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 40*x^13 + 132*x^12 + 584*x^11 - 1202*x^10 - 3900*x^9 + 8925*x^8 + 5144*x^7 - 40238*x^6 + 15732*x^5 + 140344*x^4 - 72900*x^3 - 138942*x^2 + 161516*x + 162641)
 
gp: K = bnfinit(x^16 - 4*x^15 - 40*x^13 + 132*x^12 + 584*x^11 - 1202*x^10 - 3900*x^9 + 8925*x^8 + 5144*x^7 - 40238*x^6 + 15732*x^5 + 140344*x^4 - 72900*x^3 - 138942*x^2 + 161516*x + 162641, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 40 x^{13} + 132 x^{12} + 584 x^{11} - 1202 x^{10} - 3900 x^{9} + 8925 x^{8} + 5144 x^{7} - 40238 x^{6} + 15732 x^{5} + 140344 x^{4} - 72900 x^{3} - 138942 x^{2} + 161516 x + 162641 \)

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:  \(9177836710508849627398144=2^{32}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.32$
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:  $2, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{24} a^{12} - \frac{1}{4} a^{11} - \frac{1}{24} a^{10} + \frac{1}{12} a^{9} + \frac{1}{12} a^{8} + \frac{5}{12} a^{7} + \frac{1}{8} a^{6} + \frac{1}{12} a^{5} - \frac{1}{3} a^{4} + \frac{1}{4} a^{3} + \frac{5}{24} a^{2} - \frac{1}{12} a - \frac{1}{24}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{8} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{4} a^{4} + \frac{5}{24} a^{3} + \frac{1}{6} a^{2} - \frac{1}{24} a - \frac{1}{4}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} - \frac{3}{16} a^{11} - \frac{7}{48} a^{10} - \frac{1}{24} a^{9} - \frac{5}{48} a^{8} + \frac{1}{16} a^{7} + \frac{11}{48} a^{6} + \frac{1}{3} a^{5} - \frac{7}{16} a^{4} + \frac{5}{48} a^{3} + \frac{1}{4} a^{2} + \frac{17}{48} a + \frac{17}{48}$, $\frac{1}{81825879992095417851529921947228528} a^{15} - \frac{79814223502703468830721320149575}{40912939996047708925764960973614264} a^{14} + \frac{273832145966483884930457666915805}{27275293330698472617176640649076176} a^{13} + \frac{60067712230254293485861119406105}{81825879992095417851529921947228528} a^{12} + \frac{602156729437712278559238658142288}{5114117499505963615720620121701783} a^{11} + \frac{4501748941726765477011025008940237}{27275293330698472617176640649076176} a^{10} - \frac{588961895043358795223786836741561}{27275293330698472617176640649076176} a^{9} - \frac{2105969420999930660997670071743887}{20456469998023854462882480486807132} a^{8} + \frac{1786252486118721972604030677007701}{13637646665349236308588320324538088} a^{7} + \frac{12947255448560302373145795364541047}{81825879992095417851529921947228528} a^{6} + \frac{1488881820378569417301373266103991}{81825879992095417851529921947228528} a^{5} - \frac{915114433908173257608657381140933}{13637646665349236308588320324538088} a^{4} - \frac{12135805127280586411159092067205253}{27275293330698472617176640649076176} a^{3} - \frac{8896852272647348636896655960636273}{81825879992095417851529921947228528} a^{2} + \frac{978243512350963626329970451064845}{13637646665349236308588320324538088} a - \frac{10981902357340212366352583941369627}{81825879992095417851529921947228528}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2974492304440059194155421}{248448085283941052781647139036} a^{15} - \frac{21677032949838560727539283}{331264113711921403708862852048} a^{14} + \frac{27198357424197401927130543}{331264113711921403708862852048} a^{13} - \frac{90661884276158397104923015}{165632056855960701854431426024} a^{12} + \frac{2382164314220817554368141013}{993792341135764211126588556144} a^{11} + \frac{4233466562100193936984784497}{993792341135764211126588556144} a^{10} - \frac{11304126469439559788186771195}{496896170567882105563294278072} a^{9} - \frac{21250250177299706368949263583}{993792341135764211126588556144} a^{8} + \frac{139786344512064126290363793985}{993792341135764211126588556144} a^{7} - \frac{78632020356021386650844885897}{993792341135764211126588556144} a^{6} - \frac{24607573121324748267750451937}{62112021320985263195411784759} a^{5} + \frac{645176777196702314034366286021}{993792341135764211126588556144} a^{4} + \frac{942396896679922924218445234211}{993792341135764211126588556144} a^{3} - \frac{305319732900245075399451671617}{165632056855960701854431426024} a^{2} - \frac{156916077882318414707170407279}{331264113711921403708862852048} a + \frac{1869430559056418573742730200457}{993792341135764211126588556144} \) (order $8$)
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:  \( 689983.703803 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed