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Properties

Label 16.0.12133332298...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{6}$
Root discriminant $18.00$
Ramified primes $5, 89$
Class number $1$
Class group Trivial
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -128, 64, 64, -144, 96, 0, -40, 57, -20, 0, 12, -9, 2, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 + 2*x^13 - 9*x^12 + 12*x^11 - 20*x^9 + 57*x^8 - 40*x^7 + 96*x^5 - 144*x^4 + 64*x^3 + 64*x^2 - 128*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + x^14 + 2*x^13 - 9*x^12 + 12*x^11 - 20*x^9 + 57*x^8 - 40*x^7 + 96*x^5 - 144*x^4 + 64*x^3 + 64*x^2 - 128*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + x^{14} + 2 x^{13} - 9 x^{12} + 12 x^{11} - 20 x^{9} + 57 x^{8} - 40 x^{7} + 96 x^{5} - 144 x^{4} + 64 x^{3} + 64 x^{2} - 128 x + 256 \)

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:  \(121333322988525390625=5^{12}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.00$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{8} a^{9} + \frac{7}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} + \frac{1}{16} a^{10} + \frac{7}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{9}{32} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} + \frac{1}{32} a^{11} + \frac{7}{64} a^{10} - \frac{1}{16} a^{9} + \frac{1}{4} a^{8} + \frac{3}{16} a^{7} - \frac{23}{64} a^{6} + \frac{3}{8} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{3968} a^{15} - \frac{9}{3968} a^{14} - \frac{51}{3968} a^{13} + \frac{19}{1984} a^{12} + \frac{59}{3968} a^{11} - \frac{53}{992} a^{10} + \frac{207}{992} a^{9} + \frac{323}{992} a^{8} + \frac{1625}{3968} a^{7} + \frac{11}{124} a^{6} + \frac{9}{992} a^{5} + \frac{19}{248} a^{4} + \frac{87}{248} a^{3} + \frac{57}{124} a^{2} - \frac{5}{31} a + \frac{8}{31}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{21}{496} a^{15} - \frac{17}{248} a^{14} + \frac{59}{992} a^{13} + \frac{139}{992} a^{12} - \frac{467}{992} a^{11} + \frac{161}{248} a^{10} - \frac{99}{992} a^{9} - \frac{291}{248} a^{8} + \frac{1203}{496} a^{7} - \frac{823}{496} a^{6} - \frac{689}{992} a^{5} + \frac{120}{31} a^{4} - \frac{1101}{248} a^{3} + \frac{183}{124} a^{2} + \frac{149}{62} a - \frac{82}{31} \) (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:  \( 12052.1232369 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, \(\Q(\zeta_{5})\), 4.0.11125.1, 8.0.440605625.1, 8.8.11015140625.1, 8.0.123765625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$