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Properties

Label 16.16.4292702462...3872.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{53}\cdot 383^{2}\cdot 1217^{6}$
Root discriminant $299.95$
Ramified primes $2, 383, 1217$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27809211433088, 0, -7985667216384, 0, 875091198288, 0, -46430899784, 0, 1283010849, 0, -19276476, 0, 156618, 0, -636, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 636*x^14 + 156618*x^12 - 19276476*x^10 + 1283010849*x^8 - 46430899784*x^6 + 875091198288*x^4 - 7985667216384*x^2 + 27809211433088)
 
gp: K = bnfinit(x^16 - 636*x^14 + 156618*x^12 - 19276476*x^10 + 1283010849*x^8 - 46430899784*x^6 + 875091198288*x^4 - 7985667216384*x^2 + 27809211433088, 1)
 

Normalized defining polynomial

\( x^{16} - 636 x^{14} + 156618 x^{12} - 19276476 x^{10} + 1283010849 x^{8} - 46430899784 x^{6} + 875091198288 x^{4} - 7985667216384 x^{2} + 27809211433088 \)

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:  \(4292702462281032366933505959381142863872=2^{53}\cdot 383^{2}\cdot 1217^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $299.95$
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:  $2, 383, 1217$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{9} + \frac{5}{16} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{3179519712308451965831420022903424973024} a^{14} - \frac{4676151664934996398388288511853013307}{794879928077112991457855005725856243256} a^{12} + \frac{159825351383136507351815119122192134065}{1589759856154225982915710011451712486512} a^{10} + \frac{10732739863565593432012676323492463657}{794879928077112991457855005725856243256} a^{8} - \frac{1}{4} a^{7} + \frac{410744359406572828494557458372432196281}{3179519712308451965831420022903424973024} a^{6} + \frac{178069019869966018910451737767226026405}{397439964038556495728927502862928121628} a^{4} - \frac{1}{4} a^{3} + \frac{97638831955035180581331014134129525439}{198719982019278247864463751431464060814} a^{2} + \frac{65259847387735100298246406671962}{213168088737745137815309820441337}$, $\frac{1}{6359039424616903931662840045806849946048} a^{15} - \frac{4676151664934996398388288511853013307}{1589759856154225982915710011451712486512} a^{13} - \frac{237614612655419988377112383740735987563}{3179519712308451965831420022903424973024} a^{11} + \frac{10732739863565593432012676323492463657}{1589759856154225982915710011451712486512} a^{9} - \frac{384135568670540162963297547353424046975}{6359039424616903931662840045806849946048} a^{7} - \frac{219370944168590476818475765095702095223}{794879928077112991457855005725856243256} a^{5} - \frac{101081150064243067283132737297334535375}{397439964038556495728927502862928121628} a^{3} + \frac{32629923693867550149123203335981}{213168088737745137815309820441337} a$

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:  \( 3169929188480000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1113:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1113
Character table for t16n1113 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.77888.1, 8.8.776517189632.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ $16$

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
$2$2.8.25.1$x^{8} + 8 x^{7} + 2 x^{4} + 4 x^{2} + 16 x + 2$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
2.8.28.4$x^{8} + 8 x^{5} + 16 x^{2} + 14$$8$$1$$28$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
383Data not computed
1217Data not computed