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Properties

Label 16.0.29580588602...5024.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 13^{4}\cdot 97^{2}$
Root discriminant $16.48$
Ramified primes $2, 3, 13, 97$
Class number $1$
Class group Trivial
Galois group $C_2\times D_4^2.C_2$ (as 16T509)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -256, -256, 256, 176, -128, -96, 20, 57, 10, -24, -16, 11, 8, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 8*x^13 + 11*x^12 - 16*x^11 - 24*x^10 + 10*x^9 + 57*x^8 + 20*x^7 - 96*x^6 - 128*x^5 + 176*x^4 + 256*x^3 - 256*x^2 - 256*x + 256)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 8*x^13 + 11*x^12 - 16*x^11 - 24*x^10 + 10*x^9 + 57*x^8 + 20*x^7 - 96*x^6 - 128*x^5 + 176*x^4 + 256*x^3 - 256*x^2 - 256*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 8 x^{13} + 11 x^{12} - 16 x^{11} - 24 x^{10} + 10 x^{9} + 57 x^{8} + 20 x^{7} - 96 x^{6} - 128 x^{5} + 176 x^{4} + 256 x^{3} - 256 x^{2} - 256 x + 256 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29580588602332545024=2^{24}\cdot 3^{8}\cdot 13^{4}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.48$
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:  $2, 3, 13, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} + \frac{3}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{5}{16} a^{8} + \frac{3}{8} a^{7} - \frac{3}{8} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{32} a^{13} - \frac{5}{32} a^{9} + \frac{3}{16} a^{8} + \frac{5}{16} a^{6} + \frac{13}{32} a^{5} - \frac{5}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{1088} a^{14} - \frac{3}{544} a^{13} - \frac{1}{68} a^{12} + \frac{1}{68} a^{11} - \frac{21}{1088} a^{10} + \frac{9}{272} a^{9} - \frac{125}{272} a^{8} - \frac{3}{32} a^{7} - \frac{31}{1088} a^{6} + \frac{1}{136} a^{5} + \frac{111}{272} a^{4} - \frac{9}{34} a^{3} - \frac{8}{17} a^{2} + \frac{5}{34} a + \frac{2}{17}$, $\frac{1}{2176} a^{15} + \frac{1}{136} a^{13} + \frac{7}{272} a^{12} + \frac{75}{2176} a^{11} - \frac{45}{1088} a^{10} + \frac{29}{136} a^{9} + \frac{489}{1088} a^{8} + \frac{173}{2176} a^{7} + \frac{251}{1088} a^{6} - \frac{33}{136} a^{5} - \frac{111}{272} a^{4} - \frac{21}{136} a^{3} - \frac{23}{68} a^{2} - \frac{1}{2} a + \frac{6}{17}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{329}{2176} a^{15} - \frac{69}{1088} a^{14} - \frac{415}{544} a^{13} + \frac{3}{68} a^{12} + \frac{4515}{2176} a^{11} + \frac{129}{272} a^{10} - \frac{451}{136} a^{9} - \frac{231}{64} a^{8} + \frac{9249}{2176} a^{7} + \frac{5435}{544} a^{6} - \frac{143}{272} a^{5} - \frac{6251}{272} a^{4} - \frac{235}{34} a^{3} + \frac{1171}{34} a^{2} + \frac{471}{34} a - 32 \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10144.1441049 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4^2.C_2$ (as 16T509):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$
Character table for $C_2\times D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.7488.1, 4.4.7488.1, \(\Q(\zeta_{12})\), 8.0.5438803968.1, 8.8.5438803968.1, 8.0.56070144.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97Data not computed