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Properties

Label 16.8.25023102747...2441.1
Degree $16$
Signature $[8, 4]$
Discriminant $97^{2}\cdot 277^{10}$
Root discriminant $59.55$
Ramified primes $97, 277$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.Q_8.C_6$ (as 16T732)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19531, -162807, -444159, -410800, 15628, 239050, 133161, -817, -18498, -5480, -5853, 104, 384, -168, 57, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 57*x^14 - 168*x^13 + 384*x^12 + 104*x^11 - 5853*x^10 - 5480*x^9 - 18498*x^8 - 817*x^7 + 133161*x^6 + 239050*x^5 + 15628*x^4 - 410800*x^3 - 444159*x^2 - 162807*x - 19531)
 
gp: K = bnfinit(x^16 - 5*x^15 + 57*x^14 - 168*x^13 + 384*x^12 + 104*x^11 - 5853*x^10 - 5480*x^9 - 18498*x^8 - 817*x^7 + 133161*x^6 + 239050*x^5 + 15628*x^4 - 410800*x^3 - 444159*x^2 - 162807*x - 19531, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 57 x^{14} - 168 x^{13} + 384 x^{12} + 104 x^{11} - 5853 x^{10} - 5480 x^{9} - 18498 x^{8} - 817 x^{7} + 133161 x^{6} + 239050 x^{5} + 15628 x^{4} - 410800 x^{3} - 444159 x^{2} - 162807 x - 19531 \)

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:  \(25023102747484871044535572441=97^{2}\cdot 277^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.55$
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:  $97, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2229695853437577899456561957792408206702676699} a^{15} - \frac{152111862358099408820885070148286238335154122}{2229695853437577899456561957792408206702676699} a^{14} - \frac{482415351856525411883781594448351642970271708}{2229695853437577899456561957792408206702676699} a^{13} + \frac{613028607507773602350448688039071005390439374}{2229695853437577899456561957792408206702676699} a^{12} + \frac{656696180481597655075403113034914139198979238}{2229695853437577899456561957792408206702676699} a^{11} + \frac{395394364503438016621749816037845012145293359}{2229695853437577899456561957792408206702676699} a^{10} - \frac{240548507169772966653002518177688576312314530}{2229695853437577899456561957792408206702676699} a^{9} - \frac{366783627617584946158863664903175161291034765}{2229695853437577899456561957792408206702676699} a^{8} - \frac{940522040203678782255960174090165778874415296}{2229695853437577899456561957792408206702676699} a^{7} - \frac{640037793335812851017943258996934196616503499}{2229695853437577899456561957792408206702676699} a^{6} - \frac{198725801171083737478836028984822936872237809}{2229695853437577899456561957792408206702676699} a^{5} + \frac{72187602110380940572243977516899574818666433}{2229695853437577899456561957792408206702676699} a^{4} + \frac{881650554740929076146846612657139491071622828}{2229695853437577899456561957792408206702676699} a^{3} + \frac{1021831773075866016862339602701771841333135298}{2229695853437577899456561957792408206702676699} a^{2} - \frac{1095865598464149564223266009047623447702429501}{2229695853437577899456561957792408206702676699} a + \frac{787019699367365229535035780269621652131277998}{2229695853437577899456561957792408206702676699}$

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:  \( 118054180.715 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.Q_8.C_6$ (as 16T732):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 30 conjugacy class representatives for $C_2^3.Q_8.C_6$
Character table for $C_2^3.Q_8.C_6$ is not computed

Intermediate fields

4.4.76729.1, 8.8.5887339441.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.12.0.1$x^{12} - x + 68$$1$$12$$0$$C_{12}$$[\ ]^{12}$
277Data not computed