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Properties

Label 16.0.89791815397...0896.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $17.66$
Ramified primes $2, 3, 13$
Class number $2$
Class group $[2]$
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -36, 59, -114, 270, -492, 609, -492, 270, -114, 59, -36, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 59 x^{12} - 114 x^{11} + 270 x^{10} - 492 x^{9} + 609 x^{8} - 492 x^{7} + 270 x^{6} - 114 x^{5} + 59 x^{4} - 36 x^{3} + 18 x^{2} - 6 x + 1 \)

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:  \(89791815397090000896=2^{24}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.66$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a$, $\frac{1}{2061} a^{14} - \frac{3}{229} a^{13} - \frac{332}{2061} a^{12} + \frac{31}{687} a^{11} + \frac{30}{229} a^{10} + \frac{34}{229} a^{9} - \frac{27}{229} a^{8} + \frac{290}{687} a^{7} - \frac{310}{687} a^{6} + \frac{331}{687} a^{5} - \frac{139}{687} a^{4} - \frac{66}{229} a^{3} - \frac{790}{2061} a^{2} - \frac{238}{687} a - \frac{457}{2061}$, $\frac{1}{35037} a^{15} - \frac{7}{35037} a^{14} - \frac{2933}{35037} a^{13} + \frac{1010}{35037} a^{12} + \frac{23}{11679} a^{11} + \frac{1673}{11679} a^{10} + \frac{814}{11679} a^{9} + \frac{320}{3893} a^{8} + \frac{456}{3893} a^{7} + \frac{4894}{11679} a^{6} + \frac{1214}{11679} a^{5} + \frac{686}{11679} a^{4} - \frac{6487}{35037} a^{3} + \frac{11653}{35037} a^{2} + \frac{12743}{35037} a + \frac{8722}{35037}$

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:  \( \frac{40849}{35037} a^{15} - \frac{76807}{11679} a^{14} + \frac{654586}{35037} a^{13} - \frac{415411}{11679} a^{12} + \frac{221200}{3893} a^{11} - \frac{443461}{3893} a^{10} + \frac{3223306}{11679} a^{9} - \frac{5594257}{11679} a^{8} + \frac{6416906}{11679} a^{7} - \frac{4608509}{11679} a^{6} + \frac{742459}{3893} a^{5} - \frac{280640}{3893} a^{4} + \frac{1493708}{35037} a^{3} - \frac{94212}{3893} a^{2} + \frac{433718}{35037} a - \frac{35491}{11679} \) (order $4$)
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:  \( 3239.8094211 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), 4.0.7488.4 x2, 4.2.24336.2 x2, 4.4.8112.1, 4.0.8112.1, \(\Q(i, \sqrt{13})\), 4.2.507.1, 4.2.8112.1, 8.0.65804544.2, 8.0.9475854336.4, 8.0.1052872704.3, 8.4.592240896.1, 8.0.592240896.2, 8.0.9475854336.5, 8.4.9475854336.1

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 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$