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Properties

Label 16.0.35687701699...5625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{9}$
Root discriminant $45.66$
Ramified primes $5, 61$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![797131, 3484458, 3440025, -1719225, 354385, -21421, -135868, 40415, 1735, -1875, 1477, -174, -85, -10, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 10*x^13 - 85*x^12 - 174*x^11 + 1477*x^10 - 1875*x^9 + 1735*x^8 + 40415*x^7 - 135868*x^6 - 21421*x^5 + 354385*x^4 - 1719225*x^3 + 3440025*x^2 + 3484458*x + 797131)
 
gp: K = bnfinit(x^16 - 3*x^15 + 5*x^14 - 10*x^13 - 85*x^12 - 174*x^11 + 1477*x^10 - 1875*x^9 + 1735*x^8 + 40415*x^7 - 135868*x^6 - 21421*x^5 + 354385*x^4 - 1719225*x^3 + 3440025*x^2 + 3484458*x + 797131, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 5 x^{14} - 10 x^{13} - 85 x^{12} - 174 x^{11} + 1477 x^{10} - 1875 x^{9} + 1735 x^{8} + 40415 x^{7} - 135868 x^{6} - 21421 x^{5} + 354385 x^{4} - 1719225 x^{3} + 3440025 x^{2} + 3484458 x + 797131 \)

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:  \(356877016993229400634765625=5^{15}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.66$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{2}{11} a^{12} - \frac{5}{11} a^{11} - \frac{2}{11} a^{10} - \frac{3}{11} a^{9} + \frac{3}{11} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{4}{11} a^{5} - \frac{5}{11} a^{4} + \frac{3}{11} a^{3} - \frac{5}{11} a^{2} - \frac{1}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{14} + \frac{2}{11} a^{12} - \frac{3}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{3}{11} a^{8} - \frac{4}{11} a^{7} + \frac{1}{11} a^{6} + \frac{3}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{2} - \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{27682702351368472902705412170244094718843439} a^{15} + \frac{506005726617212646594873402587567263783718}{27682702351368472902705412170244094718843439} a^{14} - \frac{638416433512917419596859025806344080345153}{27682702351368472902705412170244094718843439} a^{13} + \frac{6891708097791843953626292019650682651396397}{27682702351368472902705412170244094718843439} a^{12} + \frac{1077286550996628510703940626391115443361768}{2516609304669861172973219288204008610803949} a^{11} - \frac{4079482909806895226102188518687218871755650}{27682702351368472902705412170244094718843439} a^{10} - \frac{13034342456818351358346451711791201141958208}{27682702351368472902705412170244094718843439} a^{9} + \frac{13682601945091064611716477082889519414521691}{27682702351368472902705412170244094718843439} a^{8} + \frac{466936835182360597126792272855304237461108}{27682702351368472902705412170244094718843439} a^{7} + \frac{370717203174259910607324148266146045973015}{27682702351368472902705412170244094718843439} a^{6} - \frac{102515316182921098256197766929614932990970}{389897216216457364826836791130198517166809} a^{5} + \frac{5626189096455876671694004082070175254607244}{27682702351368472902705412170244094718843439} a^{4} - \frac{132484138939187182709957424350266343872312}{27682702351368472902705412170244094718843439} a^{3} - \frac{386930323098138855173196788940322623198291}{27682702351368472902705412170244094718843439} a^{2} - \frac{4678870053046875495343071499850836323989850}{27682702351368472902705412170244094718843439} a + \frac{12972023397719147405771947699017392442848038}{27682702351368472902705412170244094718843439}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( \frac{8922477733582279028952659918641985}{4933648610117353930262949950141524633549} a^{15} - \frac{23303232790879382305109117009823517}{4933648610117353930262949950141524633549} a^{14} + \frac{25467065567951748414313761235865154}{4933648610117353930262949950141524633549} a^{13} - \frac{102128134059627238551611294794838442}{4933648610117353930262949950141524633549} a^{12} - \frac{55816939283141068298938560802369003}{448513510010668539114813631831047693959} a^{11} - \frac{1628189500056423403802001768509111244}{4933648610117353930262949950141524633549} a^{10} + \frac{12512741811973334493783175706757459769}{4933648610117353930262949950141524633549} a^{9} - \frac{6626409110442435546330578844902202399}{4933648610117353930262949950141524633549} a^{8} + \frac{7333861731425255598695258394572976519}{4933648610117353930262949950141524633549} a^{7} + \frac{264711930437203947424420293467215995567}{4933648610117353930262949950141524633549} a^{6} - \frac{14710027004488291353897410928194632740}{69488008593202168031872534509035558219} a^{5} - \frac{247596256471180190927377355285049215553}{4933648610117353930262949950141524633549} a^{4} + \frac{2000917068001204598530506025686816685389}{4933648610117353930262949950141524633549} a^{3} - \frac{10420563123701643223109820944145780248190}{4933648610117353930262949950141524633549} a^{2} + \frac{33170924361721261807962398964472808448103}{4933648610117353930262949950141524633549} a + \frac{12828765820995170164857377162572528909123}{4933648610117353930262949950141524633549} \) (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:  \( 1835018.24291 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$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$$0$Trivial$[\ ]$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$