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Properties

Label 16.16.8325384592...0816.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{26}\cdot 3^{14}\cdot 11^{10}$
Root discriminant $36.10$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T406)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -20, -99, 236, 1268, 24, -3452, -1346, 3435, 1408, -1580, -474, 350, 56, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 33*x^14 + 56*x^13 + 350*x^12 - 474*x^11 - 1580*x^10 + 1408*x^9 + 3435*x^8 - 1346*x^7 - 3452*x^6 + 24*x^5 + 1268*x^4 + 236*x^3 - 99*x^2 - 20*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 33*x^14 + 56*x^13 + 350*x^12 - 474*x^11 - 1580*x^10 + 1408*x^9 + 3435*x^8 - 1346*x^7 - 3452*x^6 + 24*x^5 + 1268*x^4 + 236*x^3 - 99*x^2 - 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 33 x^{14} + 56 x^{13} + 350 x^{12} - 474 x^{11} - 1580 x^{10} + 1408 x^{9} + 3435 x^{8} - 1346 x^{7} - 3452 x^{6} + 24 x^{5} + 1268 x^{4} + 236 x^{3} - 99 x^{2} - 20 x + 1 \)

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:  \(8325384592016962870050816=2^{26}\cdot 3^{14}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.10$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{39} a^{14} - \frac{2}{39} a^{13} - \frac{1}{39} a^{12} + \frac{5}{39} a^{11} + \frac{2}{13} a^{10} - \frac{2}{39} a^{9} + \frac{1}{13} a^{8} + \frac{6}{13} a^{7} - \frac{6}{13} a^{6} + \frac{10}{39} a^{5} - \frac{11}{39} a^{4} - \frac{7}{39} a^{3} - \frac{7}{39} a^{2} + \frac{4}{13} a - \frac{11}{39}$, $\frac{1}{107062832808711} a^{15} - \frac{433562169242}{35687610936237} a^{14} - \frac{4362951122444}{35687610936237} a^{13} - \frac{1879194572179}{107062832808711} a^{12} - \frac{2551721749624}{35687610936237} a^{11} + \frac{1959494827832}{35687610936237} a^{10} + \frac{7482396683218}{107062832808711} a^{9} + \frac{240455446004}{2745200841249} a^{8} + \frac{14247414616874}{35687610936237} a^{7} - \frac{43089973256186}{107062832808711} a^{6} + \frac{1798004132000}{35687610936237} a^{5} - \frac{1839788318617}{35687610936237} a^{4} - \frac{9949964225803}{107062832808711} a^{3} + \frac{11934563550226}{35687610936237} a^{2} + \frac{15207071193763}{35687610936237} a + \frac{11941405308886}{107062832808711}$

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:  \( 19480556.338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T406):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

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

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.18.3$x^{8} + 14 x^{6} + 10 x^{4} + 8 x^{3} + 12 x^{2} + 20$$4$$2$$18$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$