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Properties

Label 16.4.15231185783...001.28
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 67^{6}$
Root discriminant $57.73$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28561, 0, 624962, 0, 275241, 0, -53987, 0, -16998, 0, 3847, 0, 147, 0, -43, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 43*x^14 + 147*x^12 + 3847*x^10 - 16998*x^8 - 53987*x^6 + 275241*x^4 + 624962*x^2 + 28561)
 
gp: K = bnfinit(x^16 - 43*x^14 + 147*x^12 + 3847*x^10 - 16998*x^8 - 53987*x^6 + 275241*x^4 + 624962*x^2 + 28561, 1)
 

Normalized defining polynomial

\( x^{16} - 43 x^{14} + 147 x^{12} + 3847 x^{10} - 16998 x^{8} - 53987 x^{6} + 275241 x^{4} + 624962 x^{2} + 28561 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15231185783695887615175635001=17^{14}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.73$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{26} a^{11} - \frac{3}{26} a^{9} + \frac{1}{26} a^{7} - \frac{1}{2} a^{5} + \frac{3}{13} a^{3} - \frac{1}{2} a^{2} + \frac{4}{13} a$, $\frac{1}{52} a^{12} + \frac{5}{26} a^{10} - \frac{3}{13} a^{8} - \frac{1}{4} a^{6} - \frac{7}{52} a^{4} + \frac{21}{52} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{52} a^{13} - \frac{2}{13} a^{9} + \frac{3}{52} a^{7} - \frac{7}{52} a^{5} - \frac{1}{4} a^{3} - \frac{15}{52} a - \frac{1}{2}$, $\frac{1}{34617481745400173672} a^{14} - \frac{1}{104} a^{13} - \frac{84399377017674151}{34617481745400173672} a^{12} + \frac{2740073385961267631}{17308740872700086836} a^{10} - \frac{9}{52} a^{9} + \frac{711307950930831279}{34617481745400173672} a^{8} + \frac{23}{104} a^{7} - \frac{929940032071939621}{8654370436350043418} a^{6} - \frac{19}{104} a^{5} - \frac{224383169677858389}{4327185218175021709} a^{4} + \frac{3}{8} a^{3} + \frac{6143221066487193997}{17308740872700086836} a^{2} - \frac{37}{104} a + \frac{61879473910689437}{204837170091125288}$, $\frac{1}{450027262690202257736} a^{15} - \frac{1207211093402112151}{225013631345101128868} a^{13} - \frac{1}{104} a^{12} + \frac{4071514991553582003}{225013631345101128868} a^{11} + \frac{2}{13} a^{10} - \frac{101143974876881218179}{450027262690202257736} a^{9} - \frac{7}{52} a^{8} - \frac{68627538400913084119}{450027262690202257736} a^{7} - \frac{1}{8} a^{6} - \frac{219152907470368188341}{450027262690202257736} a^{5} + \frac{33}{104} a^{4} + \frac{49899667490957269003}{450027262690202257736} a^{3} + \frac{31}{104} a^{2} - \frac{519560157664554493}{1331441605592314372} a + \frac{3}{8}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 13095545.3663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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 $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.6.123414690307499.1, 8.4.1842010303097.2, 8.2.1617217123.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed