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Properties

Label 16.0.28654123680...1824.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 3^{12}\cdot 11^{12}$
Root discriminant $60.06$
Ramified primes $2, 3, 11$
Class number $256$ (GRH)
Class group $[2, 4, 4, 8]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2500, 0, 35800, 0, 112537, 0, 116938, 0, 52027, 0, 11284, 0, 1207, 0, 58, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 58*x^14 + 1207*x^12 + 11284*x^10 + 52027*x^8 + 116938*x^6 + 112537*x^4 + 35800*x^2 + 2500)
 
gp: K = bnfinit(x^16 + 58*x^14 + 1207*x^12 + 11284*x^10 + 52027*x^8 + 116938*x^6 + 112537*x^4 + 35800*x^2 + 2500, 1)
 

Normalized defining polynomial

\( x^{16} + 58 x^{14} + 1207 x^{12} + 11284 x^{10} + 52027 x^{8} + 116938 x^{6} + 112537 x^{4} + 35800 x^{2} + 2500 \)

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:  \(28654123680257493540299341824=2^{34}\cdot 3^{12}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.06$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{7} - \frac{1}{4} a^{5} + \frac{5}{12} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{10} + \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{60} a^{11} + \frac{1}{30} a^{7} + \frac{2}{15} a^{5} - \frac{1}{2} a^{4} + \frac{1}{20} a^{3} - \frac{1}{2} a^{2} - \frac{13}{30} a$, $\frac{1}{120} a^{12} + \frac{1}{60} a^{8} + \frac{1}{15} a^{6} + \frac{11}{40} a^{4} - \frac{7}{15} a^{2} - \frac{1}{2}$, $\frac{1}{120} a^{13} + \frac{1}{60} a^{9} + \frac{1}{15} a^{7} - \frac{9}{40} a^{5} - \frac{1}{2} a^{4} + \frac{1}{30} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10511917200} a^{14} + \frac{36365213}{10511917200} a^{12} - \frac{13764853}{1751986200} a^{10} - \frac{41104153}{5255958600} a^{8} + \frac{681512117}{10511917200} a^{6} - \frac{1492517149}{3503972400} a^{4} - \frac{1}{2} a^{3} + \frac{594483479}{1313989650} a^{2} - \frac{1}{2} a - \frac{10160053}{105119172}$, $\frac{1}{105119172000} a^{15} - \frac{1}{21023834400} a^{14} - \frac{226432717}{105119172000} a^{13} + \frac{51234097}{21023834400} a^{12} - \frac{13764853}{17519862000} a^{11} + \frac{13764853}{3503972400} a^{10} - \frac{741898633}{52559586000} a^{9} - \frac{309293087}{10511917200} a^{8} - \frac{5800836823}{105119172000} a^{7} - \frac{4360683137}{21023834400} a^{6} - \frac{1755315079}{35039724000} a^{5} + \frac{1580116459}{7007944800} a^{4} + \frac{1145167303}{26279793000} a^{3} + \frac{1745609927}{5255958600} a^{2} + \frac{11983879}{210238344} a + \frac{80239501}{210238344}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $256$ (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:  \( 1040970.36274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{33}) \), 4.4.287496.1, \(\Q(\sqrt{3}, \sqrt{11})\), 4.4.1149984.1, 8.8.5289852801024.4, 8.0.427462852608.2, 8.0.3847165673472.16

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.22.98$x^{8} + 24 x^{4} + 784$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$