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Properties

Label 16.8.13096085431...1024.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{62}\cdot 7^{6}\cdot 17^{6}$
Root discriminant $88.07$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14161, 0, 545496, 0, -2196828, 0, 146872, 0, 65942, 0, -2840, 0, -508, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 508*x^12 - 2840*x^10 + 65942*x^8 + 146872*x^6 - 2196828*x^4 + 545496*x^2 + 14161)
 
gp: K = bnfinit(x^16 + 8*x^14 - 508*x^12 - 2840*x^10 + 65942*x^8 + 146872*x^6 - 2196828*x^4 + 545496*x^2 + 14161, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 508 x^{12} - 2840 x^{10} + 65942 x^{8} + 146872 x^{6} - 2196828 x^{4} + 545496 x^{2} + 14161 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13096085431976788600857793921024=2^{62}\cdot 7^{6}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.07$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{3}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{5}{16} a - \frac{5}{16}$, $\frac{1}{272} a^{10} - \frac{11}{272} a^{8} + \frac{3}{136} a^{6} - \frac{1}{136} a^{4} - \frac{3}{272} a^{2} + \frac{1}{16}$, $\frac{1}{272} a^{11} + \frac{3}{136} a^{9} - \frac{1}{16} a^{8} + \frac{3}{136} a^{7} + \frac{2}{17} a^{5} - \frac{1}{8} a^{4} + \frac{133}{272} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{16}$, $\frac{1}{544} a^{12} - \frac{13}{544} a^{8} + \frac{2}{17} a^{6} + \frac{43}{544} a^{4} + \frac{8}{17} a^{2} + \frac{9}{32}$, $\frac{1}{1088} a^{13} - \frac{1}{1088} a^{12} - \frac{1}{544} a^{11} - \frac{1}{544} a^{10} + \frac{9}{1088} a^{9} - \frac{33}{1088} a^{8} + \frac{13}{272} a^{7} - \frac{19}{272} a^{6} + \frac{47}{1088} a^{5} + \frac{97}{1088} a^{4} + \frac{131}{544} a^{3} - \frac{125}{544} a^{2} - \frac{25}{64} a + \frac{1}{64}$, $\frac{1}{49280983663572416} a^{14} + \frac{1547344227161}{2898881391974848} a^{12} + \frac{76505846300215}{49280983663572416} a^{10} + \frac{1689783559568043}{49280983663572416} a^{8} + \frac{333609805946035}{2898881391974848} a^{6} - \frac{36445570783533}{436114899677632} a^{4} + \frac{4003908416683885}{49280983663572416} a^{2} + \frac{152069107269767}{414125913139264}$, $\frac{1}{49280983663572416} a^{15} - \frac{9495084943935}{24640491831786208} a^{13} - \frac{1}{1088} a^{12} - \frac{14084197198999}{49280983663572416} a^{11} - \frac{1}{544} a^{10} + \frac{48761960457753}{12320245915893104} a^{9} + \frac{35}{1088} a^{8} + \frac{2228945048112463}{49280983663572416} a^{7} - \frac{19}{272} a^{6} + \frac{1217998279525}{218057449838816} a^{5} - \frac{39}{1088} a^{4} - \frac{7319847020717865}{49280983663572416} a^{3} - \frac{125}{544} a^{2} + \frac{19817478082821}{51765739142408} a + \frac{21}{64}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 3602563985.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.121856.1, 4.4.243712.2, 8.8.950328623104.5

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$