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Properties

Label 16.8.41816070871...0000.2
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![967184, -25661216, -1259768, 19282960, -3705440, -3321280, 1447360, -250320, -51313, 56100, -14962, 3082, -280, -120, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 120*x^13 - 280*x^12 + 3082*x^11 - 14962*x^10 + 56100*x^9 - 51313*x^8 - 250320*x^7 + 1447360*x^6 - 3321280*x^5 - 3705440*x^4 + 19282960*x^3 - 1259768*x^2 - 25661216*x + 967184)
 
gp: K = bnfinit(x^16 - 6*x^15 + 27*x^14 - 120*x^13 - 280*x^12 + 3082*x^11 - 14962*x^10 + 56100*x^9 - 51313*x^8 - 250320*x^7 + 1447360*x^6 - 3321280*x^5 - 3705440*x^4 + 19282960*x^3 - 1259768*x^2 - 25661216*x + 967184, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 27 x^{14} - 120 x^{13} - 280 x^{12} + 3082 x^{11} - 14962 x^{10} + 56100 x^{9} - 51313 x^{8} - 250320 x^{7} + 1447360 x^{6} - 3321280 x^{5} - 3705440 x^{4} + 19282960 x^{3} - 1259768 x^{2} - 25661216 x + 967184 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{56} a^{14} + \frac{1}{28} a^{13} + \frac{3}{56} a^{12} - \frac{1}{7} a^{11} + \frac{3}{14} a^{10} + \frac{13}{28} a^{9} + \frac{13}{28} a^{8} - \frac{1}{14} a^{7} - \frac{25}{56} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{2} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{1789113344139090020461382168507280916143024813230728} a^{15} + \frac{9860787566735920490088799524936371968201533950797}{1789113344139090020461382168507280916143024813230728} a^{14} + \frac{124761883832902038344883337983134325921585759743323}{1789113344139090020461382168507280916143024813230728} a^{13} + \frac{158805209228796689045499914273360345256506539720673}{1789113344139090020461382168507280916143024813230728} a^{12} - \frac{9792140356854976111399088167004666262358583314671}{894556672069545010230691084253640458071512406615364} a^{11} + \frac{190997590793978018263200445586157406176809405636501}{894556672069545010230691084253640458071512406615364} a^{10} + \frac{71647207044215112424771264541677862397223445760641}{223639168017386252557672771063410114517878101653841} a^{9} + \frac{37922055737649910154293091057038501917980921667995}{894556672069545010230691084253640458071512406615364} a^{8} - \frac{144014283401442882308159057937622062474711142233009}{1789113344139090020461382168507280916143024813230728} a^{7} + \frac{55934029784561568808234992578227032840844962644347}{255587620591298574351626024072468702306146401890104} a^{6} - \frac{35580071133553961931273759995331651884914881046995}{127793810295649287175813012036234351153073200945052} a^{5} - \frac{213773019973146861885222802383298378669837679922661}{447278336034772505115345542126820229035756203307682} a^{4} - \frac{97787444705126544219820252492805457944910881321443}{447278336034772505115345542126820229035756203307682} a^{3} + \frac{147600857138865583753204605535529881544749213951755}{447278336034772505115345542126820229035756203307682} a^{2} + \frac{61132337958038191570585580367275458028272057152235}{223639168017386252557672771063410114517878101653841} a - \frac{89572643143638948683866820273244422303852690830917}{223639168017386252557672771063410114517878101653841}$

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:  \( 174819873.824 \) (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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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