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Properties

Label 16.16.8756977801...0625.2
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 101^{2}\cdot 181^{6}$
Root discriminant $41.82$
Ramified primes $5, 101, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -79, -443, -136, 2572, 2187, -5034, -2799, 4725, 232, -1444, 209, 140, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 30*x^14 + 140*x^13 + 209*x^12 - 1444*x^11 + 232*x^10 + 4725*x^9 - 2799*x^8 - 5034*x^7 + 2187*x^6 + 2572*x^5 - 136*x^4 - 443*x^3 - 79*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 30*x^14 + 140*x^13 + 209*x^12 - 1444*x^11 + 232*x^10 + 4725*x^9 - 2799*x^8 - 5034*x^7 + 2187*x^6 + 2572*x^5 - 136*x^4 - 443*x^3 - 79*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 30 x^{14} + 140 x^{13} + 209 x^{12} - 1444 x^{11} + 232 x^{10} + 4725 x^{9} - 2799 x^{8} - 5034 x^{7} + 2187 x^{6} + 2572 x^{5} - 136 x^{4} - 443 x^{3} - 79 x^{2} + 3 x + 1 \)

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:  \(87569778018689765869140625=5^{12}\cdot 101^{2}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.82$
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:  $5, 101, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{41} a^{13} + \frac{14}{41} a^{12} - \frac{6}{41} a^{11} - \frac{13}{41} a^{10} + \frac{14}{41} a^{9} - \frac{13}{41} a^{8} + \frac{11}{41} a^{7} - \frac{2}{41} a^{6} - \frac{6}{41} a^{5} + \frac{1}{41} a^{4} - \frac{15}{41} a^{3} + \frac{2}{41} a^{2} - \frac{11}{41} a - \frac{4}{41}$, $\frac{1}{41} a^{14} + \frac{3}{41} a^{12} - \frac{11}{41} a^{11} - \frac{9}{41} a^{10} - \frac{4}{41} a^{9} - \frac{12}{41} a^{8} + \frac{8}{41} a^{7} - \frac{19}{41} a^{6} + \frac{3}{41} a^{5} + \frac{12}{41} a^{4} + \frac{7}{41} a^{3} + \frac{2}{41} a^{2} - \frac{14}{41} a + \frac{15}{41}$, $\frac{1}{217635654280529} a^{15} + \frac{584830345909}{217635654280529} a^{14} - \frac{1828749644338}{217635654280529} a^{13} + \frac{25351800457930}{217635654280529} a^{12} + \frac{80595221053698}{217635654280529} a^{11} + \frac{100773241473095}{217635654280529} a^{10} + \frac{50959785460968}{217635654280529} a^{9} + \frac{22819534764144}{217635654280529} a^{8} - \frac{96412590712465}{217635654280529} a^{7} - \frac{3667608963973}{217635654280529} a^{6} + \frac{62209650797187}{217635654280529} a^{5} - \frac{22129277792323}{217635654280529} a^{4} - \frac{67978485908976}{217635654280529} a^{3} + \frac{98139543692873}{217635654280529} a^{2} + \frac{25944719503607}{217635654280529} a + \frac{25081239055332}{217635654280529}$

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:  \( 53785926.2144 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 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

\(\Q(\sqrt{5}) \), 4.4.4525.1, 8.8.2068038125.1, 8.8.92652203125.1, 8.8.9357872515625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$