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Properties

Label 16.16.1443749059...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 3^{4}\cdot 5^{12}\cdot 11^{10}$
Root discriminant $157.57$
Ramified primes $2, 3, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T467)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9418761, 0, -185630940, 0, 129267072, 0, -32869320, 0, 3839419, 0, -236360, 0, 7978, 0, -140, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 140*x^14 + 7978*x^12 - 236360*x^10 + 3839419*x^8 - 32869320*x^6 + 129267072*x^4 - 185630940*x^2 + 9418761)
 
gp: K = bnfinit(x^16 - 140*x^14 + 7978*x^12 - 236360*x^10 + 3839419*x^8 - 32869320*x^6 + 129267072*x^4 - 185630940*x^2 + 9418761, 1)
 

Normalized defining polynomial

\( x^{16} - 140 x^{14} + 7978 x^{12} - 236360 x^{10} + 3839419 x^{8} - 32869320 x^{6} + 129267072 x^{4} - 185630940 x^{2} + 9418761 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(144374905963274060169216000000000000=2^{48}\cdot 3^{4}\cdot 5^{12}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.57$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{33} a^{10} + \frac{1}{33} a^{8} + \frac{1}{33} a^{6} - \frac{5}{33} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{33} a^{11} + \frac{1}{33} a^{9} + \frac{1}{33} a^{7} - \frac{5}{33} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{99} a^{12} + \frac{1}{99} a^{10} + \frac{1}{99} a^{8} - \frac{5}{99} a^{6} - \frac{1}{9} a^{4}$, $\frac{1}{99} a^{13} + \frac{1}{99} a^{11} + \frac{1}{99} a^{9} - \frac{5}{99} a^{7} - \frac{1}{9} a^{5}$, $\frac{1}{11171002012809153273} a^{14} - \frac{6277344467894708}{11171002012809153273} a^{12} + \frac{41817780557349988}{11171002012809153273} a^{10} + \frac{4472755812295097974}{11171002012809153273} a^{8} + \frac{3911200175198250586}{11171002012809153273} a^{6} - \frac{610979932047607523}{3723667337603051091} a^{4} - \frac{6863812171387186}{112838404169789427} a^{2} + \frac{7995359638453706}{37612801389929809}$, $\frac{1}{346301062397083751463} a^{15} + \frac{332237868041473573}{346301062397083751463} a^{13} + \frac{2749939480632296236}{346301062397083751463} a^{11} + \frac{163574905691698190044}{346301062397083751463} a^{9} + \frac{60443240664262753513}{346301062397083751463} a^{7} + \frac{42606128864981743018}{115433687465694583821} a^{5} - \frac{1624214271938368973}{3497990529263472237} a^{3} - \frac{405745455650774193}{1165996843087824079} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 1796431888770 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T467):

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 $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.17600.1, 4.4.968000.2, 4.4.22000.1, 8.8.14992384000000.4

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$