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Properties

Label 16.0.14508715371...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $136.49$
Ramified primes $2, 5, 41, 661$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T864

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![320902833165121, 0, 14564425105830, 0, -435429788627, 0, -7498846700, 0, 468553114, 0, -6666730, 0, 62812, 0, -300, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 300*x^14 + 62812*x^12 - 6666730*x^10 + 468553114*x^8 - 7498846700*x^6 - 435429788627*x^4 + 14564425105830*x^2 + 320902833165121)
 
gp: K = bnfinit(x^16 - 300*x^14 + 62812*x^12 - 6666730*x^10 + 468553114*x^8 - 7498846700*x^6 - 435429788627*x^4 + 14564425105830*x^2 + 320902833165121, 1)
 

Normalized defining polynomial

\( x^{16} - 300 x^{14} + 62812 x^{12} - 6666730 x^{10} + 468553114 x^{8} - 7498846700 x^{6} - 435429788627 x^{4} + 14564425105830 x^{2} + 320902833165121 \)

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:  \(14508715371960087713296000000000000=2^{16}\cdot 5^{12}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.49$
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:  $2, 5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{2644} a^{10} - \frac{75}{661} a^{8} + \frac{339}{1322} a^{6} + \frac{777}{2644} a^{4} - \frac{41}{2644} a^{2} - \frac{1}{2}$, $\frac{1}{2644} a^{11} - \frac{75}{661} a^{9} + \frac{339}{1322} a^{7} + \frac{777}{2644} a^{5} - \frac{41}{2644} a^{3} - \frac{1}{2} a$, $\frac{1}{214965132} a^{12} + \frac{26801}{214965132} a^{10} + \frac{80051}{1747684} a^{8} + \frac{29621509}{214965132} a^{6} - \frac{4981413}{35827522} a^{4} + \frac{385}{1983} a^{2} + \frac{1}{12}$, $\frac{1}{214965132} a^{13} + \frac{26801}{214965132} a^{11} + \frac{80051}{1747684} a^{9} + \frac{29621509}{214965132} a^{7} - \frac{4981413}{35827522} a^{5} + \frac{385}{1983} a^{3} + \frac{1}{12} a$, $\frac{1}{5560553377028506309785932904074984946972} a^{14} + \frac{1120730865668432591375623619321}{5560553377028506309785932904074984946972} a^{12} - \frac{73156480573345847707922196843545327}{926758896171417718297655484012497491162} a^{10} - \frac{94379577630904549740261640576647738707}{5560553377028506309785932904074984946972} a^{8} - \frac{230973050548516506423910884011681380695}{463379448085708859148827742006248745581} a^{6} + \frac{3507665677073185986681664118207954825}{8412334912297286399071002880597556652} a^{4} + \frac{21273342922844625166937565580795}{77601702080156510821288908901863} a^{2} + \frac{34741805907735442767168204783}{78266971336516904509620684722}$, $\frac{1}{5560553377028506309785932904074984946972} a^{15} + \frac{1120730865668432591375623619321}{5560553377028506309785932904074984946972} a^{13} - \frac{73156480573345847707922196843545327}{926758896171417718297655484012497491162} a^{11} - \frac{94379577630904549740261640576647738707}{5560553377028506309785932904074984946972} a^{9} - \frac{230973050548516506423910884011681380695}{463379448085708859148827742006248745581} a^{7} + \frac{3507665677073185986681664118207954825}{8412334912297286399071002880597556652} a^{5} + \frac{21273342922844625166937565580795}{77601702080156510821288908901863} a^{3} + \frac{34741805907735442767168204783}{78266971336516904509620684722} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( \frac{1063326857610895347440}{10998539035050410841158001044509951} a^{14} - \frac{1089462338727336445570159}{32995617105151232523474003133529853} a^{12} + \frac{236191550018005208802154966}{32995617105151232523474003133529853} a^{10} - \frac{9673756320421634864313047045}{10998539035050410841158001044509951} a^{8} + \frac{2262545976524844227821820148710}{32995617105151232523474003133529853} a^{6} - \frac{40946475944408938136923367502}{16639242110514993708257187661891} a^{4} - \frac{32715230479444885526518294}{1841914554132503639156177373} a^{2} + \frac{6077273342070251999355926}{2786557570548417003261993} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10569301888.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.26265625.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
5Data not computed
41Data not computed
661Data not computed