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Properties

Label 16.8.41816070871...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{10}\cdot 761^{5}$
Root discriminant $61.49$
Ramified primes $2, 5, 761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1360

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1114489, -2864814, -410547, 302348, -559210, 332270, -66718, -57334, 33347, -12170, 1760, 1582, -574, 84, -1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - x^14 + 84*x^13 - 574*x^12 + 1582*x^11 + 1760*x^10 - 12170*x^9 + 33347*x^8 - 57334*x^7 - 66718*x^6 + 332270*x^5 - 559210*x^4 + 302348*x^3 - 410547*x^2 - 2864814*x - 1114489)
 
gp: K = bnfinit(x^16 - 6*x^15 - x^14 + 84*x^13 - 574*x^12 + 1582*x^11 + 1760*x^10 - 12170*x^9 + 33347*x^8 - 57334*x^7 - 66718*x^6 + 332270*x^5 - 559210*x^4 + 302348*x^3 - 410547*x^2 - 2864814*x - 1114489, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - x^{14} + 84 x^{13} - 574 x^{12} + 1582 x^{11} + 1760 x^{10} - 12170 x^{9} + 33347 x^{8} - 57334 x^{7} - 66718 x^{6} + 332270 x^{5} - 559210 x^{4} + 302348 x^{3} - 410547 x^{2} - 2864814 x - 1114489 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41816070871416995840000000000=2^{24}\cdot 5^{10}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.49$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{793382615852622431922130890478139327518382736518} a^{15} + \frac{66898483036501923613365732561042562409384691594}{396691307926311215961065445239069663759191368259} a^{14} + \frac{51514522539997185634796775419314508545049644863}{793382615852622431922130890478139327518382736518} a^{13} + \frac{77674022367101099712988814949954811910569822126}{396691307926311215961065445239069663759191368259} a^{12} + \frac{362814491308690841527776782495068308349291231771}{793382615852622431922130890478139327518382736518} a^{11} - \frac{177528592631198964735636777586975127460556600361}{396691307926311215961065445239069663759191368259} a^{10} + \frac{171452436986976655348340551778124869256479685484}{396691307926311215961065445239069663759191368259} a^{9} + \frac{139084937015652481603309655546103836515362147708}{396691307926311215961065445239069663759191368259} a^{8} - \frac{66984947023023895098437851541470573133718862312}{396691307926311215961065445239069663759191368259} a^{7} - \frac{190300068431014410537261590651660878233332124698}{396691307926311215961065445239069663759191368259} a^{6} + \frac{377818714870843290297624808139693865157908104955}{793382615852622431922130890478139327518382736518} a^{5} + \frac{9642687620100370867069587649102658530270225832}{396691307926311215961065445239069663759191368259} a^{4} - \frac{261162858967493963543100588579645015144353233779}{793382615852622431922130890478139327518382736518} a^{3} - \frac{157842418554957730569327419382725227569842166917}{396691307926311215961065445239069663759191368259} a^{2} + \frac{261186840427789118772312092626583932587447044809}{793382615852622431922130890478139327518382736518} a - \frac{94815057059074196273020710439558984225828815960}{396691307926311215961065445239069663759191368259}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 180951931.535 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1360:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 65 conjugacy class representatives for t16n1360 are not computed
Character table for t16n1360 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
761Data not computed