↑

Properties

Label 16.0.15528292352...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{4}\cdot 5^{8}\cdot 61^{8}$
Root discriminant $32.50$
Ramified primes $2, 3, 5, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:C_2^2$ (as 16T119)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, 729, 0, 648, 0, 297, 0, 549, 0, -33, 0, 8, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 + 8*x^12 - 33*x^10 + 549*x^8 + 297*x^6 + 648*x^4 + 729*x^2 + 6561)
 
gp: K = bnfinit(x^16 - x^14 + 8*x^12 - 33*x^10 + 549*x^8 + 297*x^6 + 648*x^4 + 729*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} + 8 x^{12} - 33 x^{10} + 549 x^{8} + 297 x^{6} + 648 x^{4} + 729 x^{2} + 6561 \)

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:  \(1552829235277976100000000=2^{8}\cdot 3^{4}\cdot 5^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.50$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} - \frac{1}{9} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{9} - \frac{4}{27} a^{7} + \frac{2}{27} a^{5} + \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{10} - \frac{4}{81} a^{8} + \frac{2}{81} a^{6} + \frac{11}{27} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} - \frac{1}{81} a^{9} - \frac{10}{81} a^{7} + \frac{13}{27} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{75330} a^{12} - \frac{1}{162} a^{11} + \frac{61}{37665} a^{10} - \frac{1}{81} a^{9} - \frac{1717}{75330} a^{8} + \frac{11}{81} a^{7} - \frac{1687}{25110} a^{6} - \frac{5}{18} a^{5} - \frac{2003}{8370} a^{4} - \frac{5}{18} a^{3} + \frac{277}{930} a^{2} - \frac{1}{3} a - \frac{3}{310}$, $\frac{1}{75330} a^{13} - \frac{343}{75330} a^{11} + \frac{143}{75330} a^{9} - \frac{1}{18} a^{8} - \frac{1997}{25110} a^{7} - \frac{1}{9} a^{6} - \frac{206}{465} a^{5} + \frac{2}{9} a^{4} + \frac{338}{1395} a^{3} - \frac{1}{2} a^{2} - \frac{3}{310} a - \frac{1}{2}$, $\frac{1}{2937870} a^{14} - \frac{8}{1468935} a^{12} - \frac{1}{162} a^{11} - \frac{12973}{2937870} a^{10} + \frac{1}{162} a^{9} - \frac{43}{32643} a^{8} + \frac{5}{81} a^{7} - \frac{1357}{108810} a^{6} - \frac{13}{54} a^{5} + \frac{101}{390} a^{4} + \frac{1}{3} a^{3} + \frac{1679}{3627} a^{2} - \frac{1}{6} a - \frac{327}{4030}$, $\frac{1}{8813610} a^{15} - \frac{11}{1762722} a^{13} - \frac{17731}{8813610} a^{11} - \frac{1}{162} a^{10} + \frac{21031}{2937870} a^{9} + \frac{2}{81} a^{8} - \frac{9095}{97929} a^{7} - \frac{1}{81} a^{6} + \frac{45244}{163215} a^{5} - \frac{11}{54} a^{4} + \frac{54347}{108810} a^{3} - \frac{1}{18} a^{2} - \frac{48}{2015} a$

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:  $7$
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:  \( 415526.854308 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2$ (as 16T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^4:C_2^2$
Character table for $C_2^4:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), 4.0.18605.1 x2, 4.0.1525.1 x2, \(\Q(\sqrt{5}, \sqrt{61})\), 8.4.1246125690000.2, 8.0.8653650625.2, 8.4.1246125690000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$