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Properties

Label 16.0.74743809852...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 7^{4}\cdot 41^{6}$
Root discriminant $41.41$
Ramified primes $2, 5, 7, 41$
Class number $72$ (GRH)
Class group $[2, 2, 18]$ (GRH)
Galois group $C_2\wr C_2^2$ (as 16T128)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37561, 10382, 11745, -6326, 17487, -6382, 3872, -1460, 1345, -594, 260, -124, 36, -8, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 36*x^12 - 124*x^11 + 260*x^10 - 594*x^9 + 1345*x^8 - 1460*x^7 + 3872*x^6 - 6382*x^5 + 17487*x^4 - 6326*x^3 + 11745*x^2 + 10382*x + 37561)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 36*x^12 - 124*x^11 + 260*x^10 - 594*x^9 + 1345*x^8 - 1460*x^7 + 3872*x^6 - 6382*x^5 + 17487*x^4 - 6326*x^3 + 11745*x^2 + 10382*x + 37561, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} + 36 x^{12} - 124 x^{11} + 260 x^{10} - 594 x^{9} + 1345 x^{8} - 1460 x^{7} + 3872 x^{6} - 6382 x^{5} + 17487 x^{4} - 6326 x^{3} + 11745 x^{2} + 10382 x + 37561 \)

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:  \(74743809852316057600000000=2^{24}\cdot 5^{8}\cdot 7^{4}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.41$
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:  $2, 5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{54887} a^{14} - \frac{2955}{54887} a^{13} - \frac{3076}{54887} a^{12} - \frac{2479}{54887} a^{11} - \frac{6044}{54887} a^{10} - \frac{2267}{7841} a^{9} + \frac{27140}{54887} a^{8} - \frac{5039}{54887} a^{7} - \frac{25825}{54887} a^{6} + \frac{2364}{7841} a^{5} - \frac{20480}{54887} a^{4} - \frac{2863}{7841} a^{3} + \frac{21675}{54887} a^{2} - \frac{21030}{54887} a - \frac{1296}{7841}$, $\frac{1}{11293834374097242381519993223} a^{15} + \frac{74324925835345016430255}{11293834374097242381519993223} a^{14} - \frac{65737785107780789899910667}{11293834374097242381519993223} a^{13} + \frac{181336824042669544181534269}{11293834374097242381519993223} a^{12} - \frac{529654935194743558029983838}{1613404910585320340217141889} a^{11} + \frac{181763153506024047499208520}{1613404910585320340217141889} a^{10} - \frac{5087452493110020755411667294}{11293834374097242381519993223} a^{9} + \frac{98180087658154047202810309}{1613404910585320340217141889} a^{8} + \frac{5190482695775788824869171052}{11293834374097242381519993223} a^{7} - \frac{63541437319738819319967317}{11293834374097242381519993223} a^{6} + \frac{5576879346754457164323632798}{11293834374097242381519993223} a^{5} + \frac{3182038982476329593931583114}{11293834374097242381519993223} a^{4} - \frac{1012855419430769011780494082}{11293834374097242381519993223} a^{3} - \frac{573241688057230819582607808}{1613404910585320340217141889} a^{2} - \frac{1985716837522982506852196204}{11293834374097242381519993223} a + \frac{391630764319262240151915900}{1613404910585320340217141889}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{18}$, which has order $72$ (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:  \( 19190.6159055 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T128):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.65600.2, 4.4.2624.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.5143040000.4, 8.8.4303360000.1, 8.0.8645450240000.8

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

Sibling fields

Degree 8 siblings: data not computed
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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ 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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$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}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{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}$