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Properties

Label 16.0.26231448756...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{4}\cdot 41^{4}\cdot 97^{4}$
Root discriminant $33.59$
Ramified primes $2, 5, 41, 97$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^4.C_2^4$ (as 16T573)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 4096, 9728, 1536, 6976, -576, 3496, -928, 1337, -358, 471, -56, 129, -4, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 17*x^14 - 4*x^13 + 129*x^12 - 56*x^11 + 471*x^10 - 358*x^9 + 1337*x^8 - 928*x^7 + 3496*x^6 - 576*x^5 + 6976*x^4 + 1536*x^3 + 9728*x^2 + 4096*x + 4096)
 
gp: K = bnfinit(x^16 - 2*x^15 + 17*x^14 - 4*x^13 + 129*x^12 - 56*x^11 + 471*x^10 - 358*x^9 + 1337*x^8 - 928*x^7 + 3496*x^6 - 576*x^5 + 6976*x^4 + 1536*x^3 + 9728*x^2 + 4096*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 17 x^{14} - 4 x^{13} + 129 x^{12} - 56 x^{11} + 471 x^{10} - 358 x^{9} + 1337 x^{8} - 928 x^{7} + 3496 x^{6} - 576 x^{5} + 6976 x^{4} + 1536 x^{3} + 9728 x^{2} + 4096 x + 4096 \)

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:  \(2623144875606314844160000=2^{24}\cdot 5^{4}\cdot 41^{4}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.59$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{11} - \frac{3}{16} a^{9} - \frac{1}{8} a^{8} - \frac{7}{16} a^{7} - \frac{3}{8} a^{6} + \frac{7}{16} a^{5} - \frac{1}{2} a^{4} - \frac{3}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{1}{64} a^{10} - \frac{1}{16} a^{9} - \frac{15}{64} a^{8} - \frac{3}{8} a^{7} + \frac{7}{64} a^{6} - \frac{3}{32} a^{5} + \frac{9}{64} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{13} - \frac{3}{128} a^{11} - \frac{1}{64} a^{10} - \frac{23}{128} a^{9} - \frac{27}{64} a^{8} - \frac{41}{128} a^{7} + \frac{1}{16} a^{6} - \frac{3}{128} a^{5} - \frac{7}{64} a^{4} - \frac{5}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{36352} a^{14} + \frac{11}{18176} a^{13} + \frac{1}{36352} a^{12} - \frac{147}{9088} a^{11} - \frac{895}{36352} a^{10} + \frac{11}{568} a^{9} - \frac{13769}{36352} a^{8} + \frac{3217}{18176} a^{7} - \frac{6007}{36352} a^{6} + \frac{1875}{4544} a^{5} + \frac{81}{4544} a^{4} + \frac{287}{1136} a^{3} - \frac{251}{568} a^{2} - \frac{24}{71}$, $\frac{1}{155314356549059584} a^{15} - \frac{389113884957}{77657178274529792} a^{14} - \frac{65380495718631}{155314356549059584} a^{13} + \frac{198047829085333}{38828589137264896} a^{12} - \frac{2140701751665543}{155314356549059584} a^{11} + \frac{503148768266015}{9707147284316224} a^{10} - \frac{19481858681938385}{155314356549059584} a^{9} - \frac{15226450727032823}{77657178274529792} a^{8} - \frac{55646605658029903}{155314356549059584} a^{7} + \frac{7639023964442579}{19414294568632448} a^{6} + \frac{278592150819603}{1213393410539528} a^{5} - \frac{512401593906225}{2426786821079056} a^{4} + \frac{253047840815299}{1213393410539528} a^{3} - \frac{336140562099135}{1213393410539528} a^{2} + \frac{142833805195555}{303348352634882} a - \frac{3531846423526}{151674176317441}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 47378.3149221 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T573):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.6208.2, 4.4.2624.1, 4.0.254528.3, 8.0.64784502784.1, 8.4.1619612569600.2, 8.4.172134400.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$