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Properties

Label 16.8.34213459341...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{52}\cdot 3^{4}\cdot 5^{8}\cdot 7^{4}$
Root discriminant $45.54$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^4$ (as 16T459)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, -216, 0, -1620, 0, -600, 0, 1958, 0, -200, 0, -180, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 180*x^12 - 200*x^10 + 1958*x^8 - 600*x^6 - 1620*x^4 - 216*x^2 + 81)
 
gp: K = bnfinit(x^16 - 8*x^14 - 180*x^12 - 200*x^10 + 1958*x^8 - 600*x^6 - 1620*x^4 - 216*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 180 x^{12} - 200 x^{10} + 1958 x^{8} - 600 x^{6} - 1620 x^{4} - 216 x^{2} + 81 \)

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:  \(342134593410406809600000000=2^{52}\cdot 3^{4}\cdot 5^{8}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.54$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{3}{8} a$, $\frac{1}{24} a^{10} + \frac{1}{24} a^{8} + \frac{1}{6} a^{4} - \frac{1}{24} a^{2} + \frac{1}{8}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{24} a^{5} - \frac{5}{24} a^{4} + \frac{17}{48} a^{3} + \frac{19}{48} a^{2} + \frac{7}{16} a + \frac{5}{16}$, $\frac{1}{31248} a^{12} - \frac{29}{7812} a^{10} - \frac{75}{3472} a^{8} - \frac{617}{7812} a^{6} - \frac{2893}{31248} a^{4} - \frac{293}{651} a^{2} - \frac{865}{3472}$, $\frac{1}{62496} a^{13} - \frac{1}{62496} a^{12} - \frac{29}{15624} a^{11} + \frac{29}{15624} a^{10} + \frac{359}{6944} a^{9} - \frac{359}{6944} a^{8} + \frac{167}{1953} a^{7} - \frac{167}{1953} a^{6} - \frac{2893}{62496} a^{5} + \frac{2893}{62496} a^{4} + \frac{2083}{5208} a^{3} - \frac{2083}{5208} a^{2} + \frac{2173}{6944} a - \frac{2173}{6944}$, $\frac{1}{187488} a^{14} + \frac{1}{187488} a^{12} - \frac{281}{20832} a^{10} - \frac{7229}{187488} a^{8} - \frac{18229}{187488} a^{6} + \frac{10081}{62496} a^{4} - \frac{8795}{20832} a^{2} + \frac{3155}{6944}$, $\frac{1}{187488} a^{15} + \frac{1}{187488} a^{13} + \frac{51}{6944} a^{11} - \frac{1}{48} a^{10} - \frac{3323}{187488} a^{9} - \frac{1}{48} a^{8} + \frac{5207}{187488} a^{7} - \frac{1}{8} a^{6} - \frac{8147}{62496} a^{5} - \frac{5}{24} a^{4} + \frac{3791}{20832} a^{3} + \frac{19}{48} a^{2} - \frac{2487}{6944} a + \frac{5}{16}$

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:  $11$
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:  \( 30842206.659 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T459):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.179200.1, 4.4.7168.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.32112640000.1

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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$