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Properties

Label 16.8.18678870765...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{14}\cdot 1361^{5}$
Root discriminant $77.98$
Ramified primes $2, 5, 1361$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15541, -25430, -82528, 303520, -213232, -57070, -26896, 135810, -29225, -21980, 544, 2720, 383, -120, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 33*x^14 - 120*x^13 + 383*x^12 + 2720*x^11 + 544*x^10 - 21980*x^9 - 29225*x^8 + 135810*x^7 - 26896*x^6 - 57070*x^5 - 213232*x^4 + 303520*x^3 - 82528*x^2 - 25430*x + 15541)
 
gp: K = bnfinit(x^16 - 33*x^14 - 120*x^13 + 383*x^12 + 2720*x^11 + 544*x^10 - 21980*x^9 - 29225*x^8 + 135810*x^7 - 26896*x^6 - 57070*x^5 - 213232*x^4 + 303520*x^3 - 82528*x^2 - 25430*x + 15541, 1)
 

Normalized defining polynomial

\( x^{16} - 33 x^{14} - 120 x^{13} + 383 x^{12} + 2720 x^{11} + 544 x^{10} - 21980 x^{9} - 29225 x^{8} + 135810 x^{7} - 26896 x^{6} - 57070 x^{5} - 213232 x^{4} + 303520 x^{3} - 82528 x^{2} - 25430 x + 15541 \)

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:  \(1867887076585120400000000000000=2^{16}\cdot 5^{14}\cdot 1361^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.98$
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:  $2, 5, 1361$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2974636746745699530245435125329517675937071} a^{15} + \frac{192928803488765204845400531892309532728453}{2974636746745699530245435125329517675937071} a^{14} + \frac{224959152800881708053054280971807312047831}{2974636746745699530245435125329517675937071} a^{13} - \frac{949892810392701980587588332768817895482985}{2974636746745699530245435125329517675937071} a^{12} - \frac{100145566604829240685944204362198930006092}{2974636746745699530245435125329517675937071} a^{11} - \frac{1312973511928522281085219379928041158660862}{2974636746745699530245435125329517675937071} a^{10} + \frac{5885910939160408811490262033374834310981}{2974636746745699530245435125329517675937071} a^{9} - \frac{891717877774200395990451823583280718904656}{2974636746745699530245435125329517675937071} a^{8} + \frac{1065759885330604865728863757524700229589379}{2974636746745699530245435125329517675937071} a^{7} - \frac{1253684903800615718297024438438231871015890}{2974636746745699530245435125329517675937071} a^{6} + \frac{1459425575379455373896977925749195334159428}{2974636746745699530245435125329517675937071} a^{5} - \frac{1243376761303093907600078473446207402117707}{2974636746745699530245435125329517675937071} a^{4} - \frac{518496642920104210691494581055288543579848}{2974636746745699530245435125329517675937071} a^{3} + \frac{456706078772242922401785778142619983611436}{2974636746745699530245435125329517675937071} a^{2} - \frac{838792207710918931936136247327556123618137}{2974636746745699530245435125329517675937071} a + \frac{674952688940930309772418856354954058255382}{2974636746745699530245435125329517675937071}$

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:  \( 952012078.373 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.7409284000000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
1361Data not computed