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Properties

Label 16.0.92283884604...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 7841^{4}$
Root discriminant $31.46$
Ramified primes $5, 7841$
Class number $16$
Class group $[16]$
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1152731, -674471, 508254, -221070, 72849, -17144, -4558, 3719, -655, 835, 513, -142, 131, -46, 17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 17*x^14 - 46*x^13 + 131*x^12 - 142*x^11 + 513*x^10 + 835*x^9 - 655*x^8 + 3719*x^7 - 4558*x^6 - 17144*x^5 + 72849*x^4 - 221070*x^3 + 508254*x^2 - 674471*x + 1152731)
 
gp: K = bnfinit(x^16 - 3*x^15 + 17*x^14 - 46*x^13 + 131*x^12 - 142*x^11 + 513*x^10 + 835*x^9 - 655*x^8 + 3719*x^7 - 4558*x^6 - 17144*x^5 + 72849*x^4 - 221070*x^3 + 508254*x^2 - 674471*x + 1152731, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 17 x^{14} - 46 x^{13} + 131 x^{12} - 142 x^{11} + 513 x^{10} + 835 x^{9} - 655 x^{8} + 3719 x^{7} - 4558 x^{6} - 17144 x^{5} + 72849 x^{4} - 221070 x^{3} + 508254 x^{2} - 674471 x + 1152731 \)

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:  \(922838846045156494140625=5^{12}\cdot 7841^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.46$
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:  $5, 7841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31447} a^{14} - \frac{14346}{31447} a^{13} + \frac{325}{2419} a^{12} + \frac{1764}{31447} a^{11} + \frac{3332}{31447} a^{10} + \frac{2024}{31447} a^{9} - \frac{12729}{31447} a^{8} - \frac{6257}{31447} a^{7} - \frac{14732}{31447} a^{6} + \frac{11386}{31447} a^{5} - \frac{15153}{31447} a^{4} + \frac{224}{533} a^{3} + \frac{285}{2419} a^{2} - \frac{5825}{31447} a + \frac{2181}{31447}$, $\frac{1}{229982534102130884870915800409733624113} a^{15} + \frac{1726442425147594368717154460879624}{229982534102130884870915800409733624113} a^{14} + \frac{56215106714459571577592279218724445514}{229982534102130884870915800409733624113} a^{13} + \frac{5192879785739451322199963898673235787}{229982534102130884870915800409733624113} a^{12} - \frac{102921891533347173723057066915317398126}{229982534102130884870915800409733624113} a^{11} + \frac{103608400534955562740681457996866115512}{229982534102130884870915800409733624113} a^{10} - \frac{74386076489635441622491504431015164239}{229982534102130884870915800409733624113} a^{9} + \frac{17893120342002507565188300787979565823}{229982534102130884870915800409733624113} a^{8} - \frac{1646698400294379990956818066382025561}{229982534102130884870915800409733624113} a^{7} + \frac{38739487042468694129964058081249160047}{229982534102130884870915800409733624113} a^{6} - \frac{77812691210199382928618250194557102057}{229982534102130884870915800409733624113} a^{5} + \frac{78367261890663471613455348275795692436}{229982534102130884870915800409733624113} a^{4} + \frac{5032961473776254799741615260805320631}{229982534102130884870915800409733624113} a^{3} + \frac{70877497238920568040218061401904872577}{229982534102130884870915800409733624113} a^{2} - \frac{59008618781038868314900326817632209860}{229982534102130884870915800409733624113} a - \frac{22871932184656517322545478217326433479}{229982534102130884870915800409733624113}$

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

Class group and class number

$C_{16}$, which has order $16$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11170.9327013 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.980125.1, 8.0.4900625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7841Data not computed