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Properties

Label 16.0.25916019301...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}$
Root discriminant $18.87$
Ramified primes $5, 101$
Class number $2$
Class group $[2]$
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 27, -57, 83, -97, 225, -445, 641, -445, 225, -97, 83, -57, 27, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 27*x^14 - 57*x^13 + 83*x^12 - 97*x^11 + 225*x^10 - 445*x^9 + 641*x^8 - 445*x^7 + 225*x^6 - 97*x^5 + 83*x^4 - 57*x^3 + 27*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^16 - 7*x^15 + 27*x^14 - 57*x^13 + 83*x^12 - 97*x^11 + 225*x^10 - 445*x^9 + 641*x^8 - 445*x^7 + 225*x^6 - 97*x^5 + 83*x^4 - 57*x^3 + 27*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 27 x^{14} - 57 x^{13} + 83 x^{12} - 97 x^{11} + 225 x^{10} - 445 x^{9} + 641 x^{8} - 445 x^{7} + 225 x^{6} - 97 x^{5} + 83 x^{4} - 57 x^{3} + 27 x^{2} - 7 x + 1 \)

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:  \(259160193017822265625=5^{12}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.87$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{132605} a^{14} - \frac{392}{26521} a^{13} + \frac{8882}{132605} a^{12} + \frac{26612}{132605} a^{11} + \frac{25646}{132605} a^{10} - \frac{41699}{132605} a^{9} + \frac{19777}{132605} a^{8} + \frac{31349}{132605} a^{7} + \frac{46298}{132605} a^{6} - \frac{41699}{132605} a^{5} - \frac{53917}{132605} a^{4} - \frac{52951}{132605} a^{3} - \frac{8832}{26521} a^{2} - \frac{55002}{132605} a + \frac{53043}{132605}$, $\frac{1}{663025} a^{15} + \frac{2}{663025} a^{14} - \frac{8827}{132605} a^{13} + \frac{2278}{663025} a^{12} - \frac{54638}{132605} a^{11} - \frac{22432}{60275} a^{10} - \frac{162023}{663025} a^{9} + \frac{245768}{663025} a^{8} - \frac{55247}{663025} a^{7} + \frac{225762}{663025} a^{6} - \frac{235717}{663025} a^{5} + \frac{49179}{132605} a^{4} + \frac{293508}{663025} a^{3} + \frac{42479}{132605} a^{2} + \frac{291757}{663025} a - \frac{289769}{663025}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{658059}{132605} a^{15} - \frac{4376128}{132605} a^{14} + \frac{16183598}{132605} a^{13} - \frac{2863953}{12055} a^{12} + \frac{42347608}{132605} a^{11} - \frac{46664404}{132605} a^{10} + \frac{2339464}{2411} a^{9} - \frac{48903012}{26521} a^{8} + \frac{65290848}{26521} a^{7} - \frac{32088272}{26521} a^{6} + \frac{68309849}{132605} a^{5} - \frac{2769308}{12055} a^{4} + \frac{40093928}{132605} a^{3} - \frac{22007763}{132605} a^{2} + \frac{644138}{12055} a - \frac{742904}{132605} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18130.7243954 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.2525.1, 4.0.12625.1, 8.0.159390625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$