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Properties

Label 16.0.5575888916015625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 59^{2}$
Root discriminant $9.64$
Ramified primes $3, 5, 59$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2\wr C_4$ (as 16T261)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 15, -30, 49, -70, 90, -105, 111, -105, 90, -70, 49, -30, 15, -5, 1]);
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 15 x^{14} - 30 x^{13} + 49 x^{12} - 70 x^{11} + 90 x^{10} - 105 x^{9} + 111 x^{8} - 105 x^{7} + 90 x^{6} - 70 x^{5} + 49 x^{4} - 30 x^{3} + 15 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(5575888916015625=3^{8}\cdot 5^{12}\cdot 59^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.64$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 59$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29} a^{15} - \frac{6}{29} a^{14} - \frac{8}{29} a^{13} + \frac{7}{29} a^{12} + \frac{13}{29} a^{11} + \frac{4}{29} a^{10} - \frac{1}{29} a^{9} + \frac{12}{29} a^{8} + \frac{12}{29} a^{7} - \frac{1}{29} a^{6} + \frac{4}{29} a^{5} + \frac{13}{29} a^{4} + \frac{7}{29} a^{3} - \frac{8}{29} a^{2} - \frac{6}{29} a + \frac{1}{29}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{15}{29} a^{15} + \frac{32}{29} a^{14} - \frac{54}{29} a^{13} + \frac{11}{29} a^{12} - \frac{50}{29} a^{11} + \frac{114}{29} a^{10} - \frac{217}{29} a^{9} + \frac{284}{29} a^{8} - \frac{296}{29} a^{7} + \frac{276}{29} a^{6} - \frac{263}{29} a^{5} + \frac{182}{29} a^{4} - \frac{105}{29} a^{3} + \frac{4}{29} a^{2} + \frac{61}{29} a - \frac{44}{29} \) (order $30$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 155.628338249 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2\times C_2\wr C_4$ (as 16T261):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2\times C_2\wr C_4$
Character table for $C_2\times C_2\wr C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.6.74671875.1, 8.2.74671875.1, \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$