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Properties

Label 16.4.42914840847...4816.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{50}\cdot 7^{8}\cdot 137^{6}$
Root discriminant $146.06$
Ramified primes $2, 7, 137$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T1439

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 4112, 0, -37800, 0, -76944, 0, -39036, 0, -5304, 0, 144, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 144*x^12 - 5304*x^10 - 39036*x^8 - 76944*x^6 - 37800*x^4 + 4112*x^2 + 4)
 
gp: K = bnfinit(x^16 + 40*x^14 + 144*x^12 - 5304*x^10 - 39036*x^8 - 76944*x^6 - 37800*x^4 + 4112*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} + 144 x^{12} - 5304 x^{10} - 39036 x^{8} - 76944 x^{6} - 37800 x^{4} + 4112 x^{2} + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42914840847220867781580656597794816=2^{50}\cdot 7^{8}\cdot 137^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.06$
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:  $2, 7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a$, $\frac{1}{1016966974874856} a^{14} + \frac{8896284123005}{508483487437428} a^{12} + \frac{43815471456485}{508483487437428} a^{10} - \frac{6928431523613}{169494495812476} a^{8} - \frac{1}{2} a^{7} - \frac{136260598741885}{508483487437428} a^{6} - \frac{21542328553301}{127120871859357} a^{4} - \frac{109835811378511}{254241743718714} a^{2} + \frac{107765405314043}{254241743718714}$, $\frac{1}{1016966974874856} a^{15} + \frac{8896284123005}{508483487437428} a^{13} + \frac{43815471456485}{508483487437428} a^{11} - \frac{6928431523613}{169494495812476} a^{9} - \frac{136260598741885}{508483487437428} a^{7} - \frac{21542328553301}{127120871859357} a^{5} - \frac{109835811378511}{254241743718714} a^{3} + \frac{107765405314043}{254241743718714} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, 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:  $9$
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:  \( 15783967630.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1439:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1439 are not computed
Character table for t16n1439 is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.1618429490757632.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$