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Properties

Label 16.16.6894824939...7376.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 3^{12}\cdot 11^{8}\cdot 31^{4}$
Root discriminant $35.68$
Ramified primes $2, 3, 11, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_2^2$ (as 16T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-83, -586, -765, 3160, 9072, 2484, -11462, -6020, 6873, 3146, -2270, -672, 396, 62, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 33*x^14 + 62*x^13 + 396*x^12 - 672*x^11 - 2270*x^10 + 3146*x^9 + 6873*x^8 - 6020*x^7 - 11462*x^6 + 2484*x^5 + 9072*x^4 + 3160*x^3 - 765*x^2 - 586*x - 83)
 
gp: K = bnfinit(x^16 - 2*x^15 - 33*x^14 + 62*x^13 + 396*x^12 - 672*x^11 - 2270*x^10 + 3146*x^9 + 6873*x^8 - 6020*x^7 - 11462*x^6 + 2484*x^5 + 9072*x^4 + 3160*x^3 - 765*x^2 - 586*x - 83, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 33 x^{14} + 62 x^{13} + 396 x^{12} - 672 x^{11} - 2270 x^{10} + 3146 x^{9} + 6873 x^{8} - 6020 x^{7} - 11462 x^{6} + 2484 x^{5} + 9072 x^{4} + 3160 x^{3} - 765 x^{2} - 586 x - 83 \)

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:  \(6894824939562616189157376=2^{16}\cdot 3^{12}\cdot 11^{8}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.68$
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:  $2, 3, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{36121551307458} a^{15} - \frac{1334790869545}{18060775653729} a^{14} - \frac{580248412005}{12040517102486} a^{13} + \frac{2437698214501}{36121551307458} a^{12} + \frac{10396624636813}{36121551307458} a^{11} + \frac{5183009276015}{18060775653729} a^{10} + \frac{1947459747009}{6020258551243} a^{9} + \frac{13889708264165}{36121551307458} a^{8} + \frac{48609760664}{18060775653729} a^{7} + \frac{7293711879863}{36121551307458} a^{6} - \frac{5316593385155}{12040517102486} a^{5} + \frac{11689935975179}{36121551307458} a^{4} - \frac{2969624769709}{12040517102486} a^{3} + \frac{1450697850945}{6020258551243} a^{2} - \frac{2473307580547}{12040517102486} a - \frac{13283698572643}{36121551307458}$

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:  \( 16492226.571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{3}) \), 4.4.4752.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 4.4.13068.1 x2, 8.8.21700825344.1, 8.8.2625799866624.3, 8.8.2732361984.1

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

Sibling fields

Degree 8 siblings: data not computed
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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$