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Properties

Label 16.16.1240729860...9728.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{66}\cdot 113^{3}\cdot 1039^{4}$
Root discriminant $240.36$
Ramified primes $2, 113, 1039$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T859

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![131686311606833, 0, -35156129966736, 0, 3219023669900, 0, -133415695888, 0, 2901858458, 0, -35068304, 0, 233444, 0, -784, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 784*x^14 + 233444*x^12 - 35068304*x^10 + 2901858458*x^8 - 133415695888*x^6 + 3219023669900*x^4 - 35156129966736*x^2 + 131686311606833)
 
gp: K = bnfinit(x^16 - 784*x^14 + 233444*x^12 - 35068304*x^10 + 2901858458*x^8 - 133415695888*x^6 + 3219023669900*x^4 - 35156129966736*x^2 + 131686311606833, 1)
 

Normalized defining polynomial

\( x^{16} - 784 x^{14} + 233444 x^{12} - 35068304 x^{10} + 2901858458 x^{8} - 133415695888 x^{6} + 3219023669900 x^{4} - 35156129966736 x^{2} + 131686311606833 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(124072986059627688411885857786694729728=2^{66}\cdot 113^{3}\cdot 1039^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.36$
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:  $2, 113, 1039$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4156} a^{10} + \frac{255}{4156} a^{8} + \frac{177}{1039} a^{6} + \frac{6}{1039} a^{4} - \frac{1007}{4156} a^{2} + \frac{1}{4}$, $\frac{1}{4156} a^{11} + \frac{255}{4156} a^{9} + \frac{177}{1039} a^{7} + \frac{6}{1039} a^{5} - \frac{1007}{4156} a^{3} + \frac{1}{4} a$, $\frac{1}{4318084} a^{12} + \frac{255}{4318084} a^{10} - \frac{145283}{1079521} a^{8} + \frac{52995}{1079521} a^{6} - \frac{948575}{4318084} a^{4} - \frac{867}{4156} a^{2}$, $\frac{1}{4318084} a^{13} + \frac{255}{4318084} a^{11} + \frac{498389}{4318084} a^{9} - \frac{1}{4} a^{8} - \frac{973531}{2159042} a^{7} - \frac{1}{2} a^{6} + \frac{1210467}{4318084} a^{5} - \frac{1}{2} a^{4} + \frac{1211}{4156} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{615789707227455893476039030052262628} a^{14} - \frac{60197215049786034952240310209}{615789707227455893476039030052262628} a^{12} - \frac{23431212663711120701824618206097}{615789707227455893476039030052262628} a^{10} + \frac{138449003468459132823091176767931989}{615789707227455893476039030052262628} a^{8} + \frac{210673497676556571715890455355617561}{615789707227455893476039030052262628} a^{6} + \frac{184862309223521875043674122652957}{592675367880130792565966342687452} a^{4} + \frac{64220935988209095435638258859}{570428650510231754153961831268} a^{2} + \frac{22435297427867574974356539}{549016987979048849041349212}$, $\frac{1}{615789707227455893476039030052262628} a^{15} - \frac{60197215049786034952240310209}{615789707227455893476039030052262628} a^{13} - \frac{23431212663711120701824618206097}{615789707227455893476039030052262628} a^{11} - \frac{3874605834601210136479645186283417}{153947426806863973369009757513065657} a^{9} - \frac{1}{4} a^{8} - \frac{97221355937171375022129059670513753}{615789707227455893476039030052262628} a^{7} - \frac{1}{2} a^{6} - \frac{111475374716543521239309048690769}{592675367880130792565966342687452} a^{5} - \frac{1}{2} a^{4} - \frac{220993389266906781641342656775}{570428650510231754153961831268} a^{3} - \frac{1}{2} a^{2} + \frac{79844772211314893617346921}{274508493989524424520674606} a + \frac{1}{4}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 121969797868000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T859:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n859
Character table for t16n859 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7583301632.1

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
Arithmetically equvalently 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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$113$113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1039Data not computed