↑

Properties

Label 16.0.79230447076...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{6}\cdot 5^{8}\cdot 1009^{4}$
Root discriminant $64.00$
Ramified primes $2, 3, 5, 1009$
Class number $440$ (GRH)
Class group $[2, 220]$ (GRH)
Galois group 16T1701

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1089, 0, 7587, 0, 16797, 0, 17061, 0, 9014, 0, 2595, 0, 405, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 405*x^12 + 2595*x^10 + 9014*x^8 + 17061*x^6 + 16797*x^4 + 7587*x^2 + 1089)
 
gp: K = bnfinit(x^16 + 32*x^14 + 405*x^12 + 2595*x^10 + 9014*x^8 + 17061*x^6 + 16797*x^4 + 7587*x^2 + 1089, 1)
 

Normalized defining polynomial

\( x^{16} + 32 x^{14} + 405 x^{12} + 2595 x^{10} + 9014 x^{8} + 17061 x^{6} + 16797 x^{4} + 7587 x^{2} + 1089 \)

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:  \(79230447076976256614400000000=2^{28}\cdot 3^{6}\cdot 5^{8}\cdot 1009^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.00$
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:  $2, 3, 5, 1009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{4}{15} a^{10} + \frac{2}{5} a^{6} - \frac{7}{15} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{165} a^{13} + \frac{41}{165} a^{11} + \frac{1}{11} a^{9} - \frac{3}{55} a^{7} + \frac{23}{165} a^{5} + \frac{3}{55} a^{3} + \frac{27}{55} a$, $\frac{1}{4909245} a^{14} + \frac{87293}{4909245} a^{12} - \frac{297171}{1636415} a^{10} - \frac{478393}{1636415} a^{8} - \frac{61151}{981849} a^{6} + \frac{150562}{327283} a^{4} - \frac{302902}{1636415} a^{2} - \frac{43736}{148765}$, $\frac{1}{4909245} a^{15} - \frac{1966}{4909245} a^{13} + \frac{119371}{1636415} a^{11} + \frac{711727}{1636415} a^{9} + \frac{497576}{4909245} a^{7} + \frac{68491}{1636415} a^{5} - \frac{570679}{1636415} a^{3} + \frac{381741}{1636415} a$

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

Class group and class number

$C_{2}\times C_{220}$, which has order $440$ (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:  \( 223881.856588 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1701:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 119 conjugacy class representatives for t16n1701 are not computed
Character table for t16n1701 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.25225.1, 8.8.366509160000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.20.44$x^{8} + 12 x^{7} + 24 x^{4} + 8 x^{3} + 48 x^{2} + 60$$4$$2$$20$$((C_8 : C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1009Data not computed