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Properties

Label 16.0.84225235721...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 53^{14}$
Root discriminant $131.93$
Ramified primes $5, 53$
Class number $5618$ (GRH)
Class group $[53, 106]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157623296, 529353216, 836081984, 585873856, 279233036, 91551908, 16657011, 4400188, 1930997, 278072, 23901, 18622, 1631, -66, 109, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 109*x^14 - 66*x^13 + 1631*x^12 + 18622*x^11 + 23901*x^10 + 278072*x^9 + 1930997*x^8 + 4400188*x^7 + 16657011*x^6 + 91551908*x^5 + 279233036*x^4 + 585873856*x^3 + 836081984*x^2 + 529353216*x + 157623296)
 
gp: K = bnfinit(x^16 - 6*x^15 + 109*x^14 - 66*x^13 + 1631*x^12 + 18622*x^11 + 23901*x^10 + 278072*x^9 + 1930997*x^8 + 4400188*x^7 + 16657011*x^6 + 91551908*x^5 + 279233036*x^4 + 585873856*x^3 + 836081984*x^2 + 529353216*x + 157623296, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 109 x^{14} - 66 x^{13} + 1631 x^{12} + 18622 x^{11} + 23901 x^{10} + 278072 x^{9} + 1930997 x^{8} + 4400188 x^{7} + 16657011 x^{6} + 91551908 x^{5} + 279233036 x^{4} + 585873856 x^{3} + 836081984 x^{2} + 529353216 x + 157623296 \)

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:  \(8422523572126121633218804931640625=5^{14}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.93$
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:  $5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{8} a^{8} + \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{24} a^{4} - \frac{11}{24} a^{3} + \frac{7}{24} a^{2} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{16} a^{9} + \frac{1}{24} a^{8} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} + \frac{5}{48} a^{5} - \frac{5}{48} a^{4} - \frac{23}{48} a^{3} + \frac{1}{4} a^{2} - \frac{1}{12} a$, $\frac{1}{48} a^{12} + \frac{1}{48} a^{9} - \frac{1}{24} a^{7} + \frac{1}{48} a^{6} - \frac{1}{24} a^{5} - \frac{1}{24} a^{4} - \frac{7}{48} a^{3} - \frac{3}{8} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{192} a^{13} + \frac{1}{192} a^{11} + \frac{1}{64} a^{9} + \frac{5}{192} a^{7} - \frac{5}{96} a^{6} + \frac{7}{64} a^{5} + \frac{5}{32} a^{4} + \frac{19}{192} a^{3} + \frac{43}{96} a^{2} + \frac{5}{24} a - \frac{1}{2}$, $\frac{1}{57216} a^{14} - \frac{17}{9536} a^{13} - \frac{113}{19072} a^{12} + \frac{97}{9536} a^{11} + \frac{847}{57216} a^{10} - \frac{193}{28608} a^{9} + \frac{2261}{57216} a^{8} - \frac{141}{2384} a^{7} + \frac{455}{19072} a^{6} + \frac{1103}{4768} a^{5} - \frac{4957}{57216} a^{4} - \frac{3791}{14304} a^{3} - \frac{6619}{14304} a^{2} - \frac{65}{3576} a - \frac{27}{298}$, $\frac{1}{4027813698328014668056948477057814788602192312288768} a^{15} - \frac{16955072650023459644984804826692925055024198733}{2013906849164007334028474238528907394301096156144384} a^{14} + \frac{263215193555206474451647197185785586806044779887}{1342604566109338222685649492352604929534064104096256} a^{13} + \frac{3610154721936573738484269534349099512734140773519}{671302283054669111342824746176302464767032052048128} a^{12} - \frac{10864044568917990270573146398162158187080983482107}{1342604566109338222685649492352604929534064104096256} a^{11} + \frac{34130798291590299987319605035607576229468325084665}{2013906849164007334028474238528907394301096156144384} a^{10} + \frac{2393795519238094273881889091968818056621986788415}{1342604566109338222685649492352604929534064104096256} a^{9} - \frac{7240303423416346793565423430843636160907376932931}{1006953424582003667014237119264453697150548078072192} a^{8} + \frac{53448406961809892731994132208616318072492740139933}{4027813698328014668056948477057814788602192312288768} a^{7} - \frac{1814242680950291343481856648903070500044146426513}{167825570763667277835706186544075616191758013012032} a^{6} + \frac{791016257234469842848949488150161231162450243058219}{4027813698328014668056948477057814788602192312288768} a^{5} + \frac{41665759158125414063292362043133887155450571984381}{167825570763667277835706186544075616191758013012032} a^{4} + \frac{271815066250794207916452696980859998911209976669597}{1006953424582003667014237119264453697150548078072192} a^{3} - \frac{3556943525853040704989953046783828452774664670461}{20978196345458409729463273318009452023969751626504} a^{2} - \frac{1001030942818295704654206480872765534753981029}{22901961075827958219938071307870580812194051994} a + \frac{5856338439165397876053770261375526358477167066785}{15733647259093807297097454988507089017977313719878}$

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

Class group and class number

$C_{53}\times C_{106}$, which has order $5618$ (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:  \( 17392693.3571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{265}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{5}, \sqrt{53})\), 4.4.18609625.2, 4.4.18609625.1, 8.8.346318142640625.1, 8.0.91774307799765625.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
53Data not computed