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Properties

Label 16.4.66900227008...9293.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{8}\cdot 13^{9}\cdot 1327^{8}$
Root discriminant $267.05$
Ramified primes $3, 13, 1327$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group 16T1675

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![205891711, -125645924, 232361572, -184526648, 71298922, -23633591, 8525768, -23935, -210018, 46333, -49354, 4150, -202, 31, 57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 57*x^14 + 31*x^13 - 202*x^12 + 4150*x^11 - 49354*x^10 + 46333*x^9 - 210018*x^8 - 23935*x^7 + 8525768*x^6 - 23633591*x^5 + 71298922*x^4 - 184526648*x^3 + 232361572*x^2 - 125645924*x + 205891711)
 
gp: K = bnfinit(x^16 - 2*x^15 + 57*x^14 + 31*x^13 - 202*x^12 + 4150*x^11 - 49354*x^10 + 46333*x^9 - 210018*x^8 - 23935*x^7 + 8525768*x^6 - 23633591*x^5 + 71298922*x^4 - 184526648*x^3 + 232361572*x^2 - 125645924*x + 205891711, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 57 x^{14} + 31 x^{13} - 202 x^{12} + 4150 x^{11} - 49354 x^{10} + 46333 x^{9} - 210018 x^{8} - 23935 x^{7} + 8525768 x^{6} - 23633591 x^{5} + 71298922 x^{4} - 184526648 x^{3} + 232361572 x^{2} - 125645924 x + 205891711 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(669002270083080728394108412888785769293=3^{8}\cdot 13^{9}\cdot 1327^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $267.05$
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:  $3, 13, 1327$
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}{1272393005179858853974946383393566358568541257038870458388609463} a^{15} - \frac{173628916668799856633108655679785647670210609647920425414328904}{1272393005179858853974946383393566358568541257038870458388609463} a^{14} - \frac{165076139578387895803238514274466117844767502705992421088030562}{1272393005179858853974946383393566358568541257038870458388609463} a^{13} + \frac{135360037088702748106785551892786723012984196137577413512583148}{1272393005179858853974946383393566358568541257038870458388609463} a^{12} + \frac{498688343489062415662572862159510501183085186663938216319293396}{1272393005179858853974946383393566358568541257038870458388609463} a^{11} + \frac{88769038583911142945196543188439769222487820338763761999045269}{1272393005179858853974946383393566358568541257038870458388609463} a^{10} + \frac{90183858127299840355279027543907344802326508700700557873523133}{1272393005179858853974946383393566358568541257038870458388609463} a^{9} + \frac{528496883421778536199357191450783240106916502328651263823272102}{1272393005179858853974946383393566358568541257038870458388609463} a^{8} + \frac{435150339314099287490870558956400102580093575414837187344864392}{1272393005179858853974946383393566358568541257038870458388609463} a^{7} + \frac{526439900284196818980544827441407059908916161383583302369156271}{1272393005179858853974946383393566358568541257038870458388609463} a^{6} - \frac{132409446437804893268910317168183318228299434281249849190577103}{1272393005179858853974946383393566358568541257038870458388609463} a^{5} - \frac{179769101048250134290701317651924271856533918081435273840046700}{1272393005179858853974946383393566358568541257038870458388609463} a^{4} - \frac{589756673885707665892986655985362245628526766005281523533616199}{1272393005179858853974946383393566358568541257038870458388609463} a^{3} - \frac{20143440204681780644929352864488952130182709106509451548562633}{1272393005179858853974946383393566358568541257038870458388609463} a^{2} + \frac{342398988391143547677570666722256751074041771840135019057580794}{1272393005179858853974946383393566358568541257038870458388609463} a - \frac{582961611800040073344232665540286109255033502744283862351853169}{1272393005179858853974946383393566358568541257038870458388609463}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  $9$
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:  \( 146882674701 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1327Data not computed