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Properties

Label 16.0.22294549101...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 5^{12}\cdot 11^{4}\cdot 29^{6}$
Root discriminant $51.20$
Ramified primes $2, 5, 11, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![197009296, 0, -113326664, 0, 32389280, 0, -6088896, 0, 777889, 0, -61956, 0, 2940, 0, -79, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 79*x^14 + 2940*x^12 - 61956*x^10 + 777889*x^8 - 6088896*x^6 + 32389280*x^4 - 113326664*x^2 + 197009296)
 
gp: K = bnfinit(x^16 - 79*x^14 + 2940*x^12 - 61956*x^10 + 777889*x^8 - 6088896*x^6 + 32389280*x^4 - 113326664*x^2 + 197009296, 1)
 

Normalized defining polynomial

\( x^{16} - 79 x^{14} + 2940 x^{12} - 61956 x^{10} + 777889 x^{8} - 6088896 x^{6} + 32389280 x^{4} - 113326664 x^{2} + 197009296 \)

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:  \(2229454910146816000000000000=2^{20}\cdot 5^{12}\cdot 11^{4}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.20$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} - \frac{1}{11} a^{6} + \frac{2}{11} a^{4} - \frac{2}{11} a^{2}$, $\frac{1}{11} a^{9} - \frac{1}{11} a^{7} + \frac{2}{11} a^{5} - \frac{2}{11} a^{3}$, $\frac{1}{22} a^{10} - \frac{1}{22} a^{8} + \frac{1}{11} a^{6} - \frac{1}{11} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{22} a^{11} - \frac{1}{22} a^{9} + \frac{1}{11} a^{7} - \frac{1}{11} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{484} a^{12} + \frac{9}{484} a^{10} - \frac{2}{121} a^{8} + \frac{32}{121} a^{6} + \frac{13}{484} a^{4} + \frac{5}{11} a^{2}$, $\frac{1}{484} a^{13} + \frac{9}{484} a^{11} - \frac{2}{121} a^{9} + \frac{32}{121} a^{7} + \frac{13}{484} a^{5} + \frac{5}{11} a^{3}$, $\frac{1}{479049551782455894739048} a^{14} + \frac{11473068556090094633}{479049551782455894739048} a^{12} + \frac{1024495785748407615609}{59881193972806986842381} a^{10} - \frac{1641821972425074718691}{59881193972806986842381} a^{8} - \frac{144528153031091838130175}{479049551782455894739048} a^{6} - \frac{2217986790958418893813}{5443744906618816985671} a^{4} + \frac{476241374420048468831}{989771801203421270122} a^{2} + \frac{730110668061288368}{1551366459566491019}$, $\frac{1}{479049551782455894739048} a^{15} + \frac{11473068556090094633}{479049551782455894739048} a^{13} + \frac{1024495785748407615609}{59881193972806986842381} a^{11} - \frac{1641821972425074718691}{59881193972806986842381} a^{9} - \frac{144528153031091838130175}{479049551782455894739048} a^{7} - \frac{2217986790958418893813}{5443744906618816985671} a^{5} + \frac{476241374420048468831}{989771801203421270122} a^{3} + \frac{730110668061288368}{1551366459566491019} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{1130420807884223}{15453211347821157894808} a^{14} + \frac{86195971412214649}{15453211347821157894808} a^{12} - \frac{779515291892782295}{3863302836955289473702} a^{10} + \frac{15780225892044628463}{3863302836955289473702} a^{8} - \frac{752355711174331246231}{15453211347821157894808} a^{6} + \frac{61967616775622678974}{175604674407058612441} a^{4} - \frac{27560223353599799739}{15964061309732601131} a^{2} + \frac{247802539841185442}{50044079340854549} \) (order $10$)
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:  \( 15103860.135 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.3625.1, 4.4.725.1, 8.0.13140625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.4$x^{8} + 2 x^{6} + 16$$2$$4$$12$$C_2^3: C_4$$[2, 2, 3]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$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}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$