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Properties

Label 16.8.18682108719...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $138.66$
Ramified primes $2, 5, 61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T864

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3269009959, 4442079620, 3833770698, -3941427530, 267067685, 206994040, -13381432, 11869340, -3314811, -99060, -61068, -4960, 2925, -10, 102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 102*x^14 - 10*x^13 + 2925*x^12 - 4960*x^11 - 61068*x^10 - 99060*x^9 - 3314811*x^8 + 11869340*x^7 - 13381432*x^6 + 206994040*x^5 + 267067685*x^4 - 3941427530*x^3 + 3833770698*x^2 + 4442079620*x - 3269009959)
 
gp: K = bnfinit(x^16 + 102*x^14 - 10*x^13 + 2925*x^12 - 4960*x^11 - 61068*x^10 - 99060*x^9 - 3314811*x^8 + 11869340*x^7 - 13381432*x^6 + 206994040*x^5 + 267067685*x^4 - 3941427530*x^3 + 3833770698*x^2 + 4442079620*x - 3269009959, 1)
 

Normalized defining polynomial

\( x^{16} + 102 x^{14} - 10 x^{13} + 2925 x^{12} - 4960 x^{11} - 61068 x^{10} - 99060 x^{9} - 3314811 x^{8} + 11869340 x^{7} - 13381432 x^{6} + 206994040 x^{5} + 267067685 x^{4} - 3941427530 x^{3} + 3833770698 x^{2} + 4442079620 x - 3269009959 \)

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:  \(18682108719227553550336000000000000=2^{24}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.66$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{9} + \frac{4}{9} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{2}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a$, $\frac{1}{9} a^{13} + \frac{4}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{1053} a^{14} - \frac{2}{81} a^{13} - \frac{7}{351} a^{12} + \frac{16}{1053} a^{11} + \frac{100}{1053} a^{10} - \frac{103}{1053} a^{9} + \frac{355}{1053} a^{8} + \frac{215}{1053} a^{7} - \frac{95}{1053} a^{6} + \frac{25}{1053} a^{5} + \frac{233}{1053} a^{4} + \frac{38}{1053} a^{3} - \frac{97}{1053} a^{2} + \frac{148}{1053} a - \frac{64}{1053}$, $\frac{1}{940710189634981187698766777766090715006592542135138612707792858295317} a^{15} + \frac{180551770654140128686555418280399193249168277118166238110139501357}{940710189634981187698766777766090715006592542135138612707792858295317} a^{14} - \frac{4463549897700450632259162870574105371060020840346558186558960724781}{940710189634981187698766777766090715006592542135138612707792858295317} a^{13} - \frac{46688426583857291833242127581538836521902764156332397100229467066166}{940710189634981187698766777766090715006592542135138612707792858295317} a^{12} - \frac{45808441760242152422883916091976664762593135732059763327264657263941}{940710189634981187698766777766090715006592542135138612707792858295317} a^{11} + \frac{15521811383224175703369740458870205879252050400575405455126682356617}{34841118134628932877732102880225582037281205264264393063251587344271} a^{10} + \frac{92378296155676260905889771695525098525520981371634884523288886400922}{313570063211660395899588925922030238335530847378379537569264286098439} a^{9} + \frac{42051148282035024120361439497143269426704626893996227771236543598499}{313570063211660395899588925922030238335530847378379537569264286098439} a^{8} - \frac{16755674121141669887799136061667370998484950084701847349911378548543}{104523354403886798633196308640676746111843615792793179189754762032813} a^{7} + \frac{175912392585666777476175251190169067228284969455910866901303181248238}{940710189634981187698766777766090715006592542135138612707792858295317} a^{6} - \frac{8384030926311336413986678372197582073502436882208315190886417022754}{104523354403886798633196308640676746111843615792793179189754762032813} a^{5} - \frac{377791548527573209574035709296533732186698067186326458260883809804338}{940710189634981187698766777766090715006592542135138612707792858295317} a^{4} - \frac{90054927554528185673588417650497930180456992887023823546916078782130}{940710189634981187698766777766090715006592542135138612707792858295317} a^{3} + \frac{28620143467029671174411870645317763344400836154348006559306889720336}{313570063211660395899588925922030238335530847378379537569264286098439} a^{2} + \frac{11013948681255332472695235487281686541685822305388863990226087406267}{24120774093204645838429917378617710641194680567567656736097252776803} a + \frac{241972455421844338937532897326933520593824108500647743986526600599170}{940710189634981187698766777766090715006592542135138612707792858295317}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 77987161152.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
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}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$