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Properties

Label 16.0.24885365962...8321.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 41^{14}$
Root discriminant $44.64$
Ramified primes $3, 41$
Class number $17$ (GRH)
Class group $[17]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3481, 9440, 16573, 18226, 12333, -6620, -1809, 6119, 1681, -795, 348, 108, -46, 19, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + 19*x^13 - 46*x^12 + 108*x^11 + 348*x^10 - 795*x^9 + 1681*x^8 + 6119*x^7 - 1809*x^6 - 6620*x^5 + 12333*x^4 + 18226*x^3 + 16573*x^2 + 9440*x + 3481)
 
gp: K = bnfinit(x^16 - x^15 - 2*x^14 + 19*x^13 - 46*x^12 + 108*x^11 + 348*x^10 - 795*x^9 + 1681*x^8 + 6119*x^7 - 1809*x^6 - 6620*x^5 + 12333*x^4 + 18226*x^3 + 16573*x^2 + 9440*x + 3481, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 2 x^{14} + 19 x^{13} - 46 x^{12} + 108 x^{11} + 348 x^{10} - 795 x^{9} + 1681 x^{8} + 6119 x^{7} - 1809 x^{6} - 6620 x^{5} + 12333 x^{4} + 18226 x^{3} + 16573 x^{2} + 9440 x + 3481 \)

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:  \(248853659625840981300978321=3^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.64$
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:  $3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(123=3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{123}(1,·)$, $\chi_{123}(68,·)$, $\chi_{123}(73,·)$, $\chi_{123}(14,·)$, $\chi_{123}(79,·)$, $\chi_{123}(83,·)$, $\chi_{123}(85,·)$, $\chi_{123}(91,·)$, $\chi_{123}(32,·)$, $\chi_{123}(38,·)$, $\chi_{123}(40,·)$, $\chi_{123}(44,·)$, $\chi_{123}(109,·)$, $\chi_{123}(50,·)$, $\chi_{123}(55,·)$, $\chi_{123}(122,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{1707476} a^{14} + \frac{192153}{1707476} a^{13} + \frac{234713}{1707476} a^{12} - \frac{71661}{853738} a^{11} + \frac{287449}{1707476} a^{10} - \frac{55570}{426869} a^{9} - \frac{626399}{1707476} a^{8} + \frac{102131}{1707476} a^{7} + \frac{77553}{853738} a^{6} - \frac{76351}{426869} a^{5} + \frac{113511}{1707476} a^{4} - \frac{127560}{426869} a^{3} - \frac{118616}{426869} a^{2} - \frac{50739}{426869} a - \frac{595647}{1707476}$, $\frac{1}{2186323211829147079065695608} a^{15} - \frac{72631527650803483524}{273290401478643384883211951} a^{14} - \frac{54737472532448640844023789}{1093161605914573539532847804} a^{13} - \frac{544205082222286913692202411}{2186323211829147079065695608} a^{12} + \frac{287567327684082199995096215}{2186323211829147079065695608} a^{11} + \frac{44247369540234300850023171}{2186323211829147079065695608} a^{10} + \frac{357310628646521741507869183}{2186323211829147079065695608} a^{9} - \frac{110444051752657555586851427}{273290401478643384883211951} a^{8} - \frac{613573665333642643488797675}{2186323211829147079065695608} a^{7} + \frac{63129577163923251512075274}{273290401478643384883211951} a^{6} + \frac{528075951947141227110669675}{2186323211829147079065695608} a^{5} + \frac{283081699311411067849803623}{2186323211829147079065695608} a^{4} - \frac{64717484780665901203806235}{273290401478643384883211951} a^{3} + \frac{306431900315269144488003341}{1093161605914573539532847804} a^{2} - \frac{115716435589861295522562189}{2186323211829147079065695608} a + \frac{3373884461101995363724165}{37056325624222831848571112}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  \( \frac{894639924229478243}{5121766190164071598232} a^{15} - \frac{152751242075448640}{640220773770508949779} a^{14} - \frac{646766157832868129}{2560883095082035799116} a^{13} + \frac{17717945834192457955}{5121766190164071598232} a^{12} - \frac{48644474209507293499}{5121766190164071598232} a^{11} + \frac{117092762148898392873}{5121766190164071598232} a^{10} + \frac{268523192070573139009}{5121766190164071598232} a^{9} - \frac{102907042458411954437}{640220773770508949779} a^{8} + \frac{1898943894331130006287}{5121766190164071598232} a^{7} + \frac{585523487795936891095}{640220773770508949779} a^{6} - \frac{3376211972516412704827}{5121766190164071598232} a^{5} - \frac{3488266906956270639015}{5121766190164071598232} a^{4} + \frac{3026771679843962481695}{1280441547541017899558} a^{3} + \frac{5618592613120417020773}{2560883095082035799116} a^{2} + \frac{14235277675575053691593}{5121766190164071598232} a + \frac{133453547170422699571}{86809596443458840648} \) (order $6$)
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:  \( 541284.897174 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-123}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{41})\), 4.4.68921.1, 4.0.620289.1, 8.0.384758443521.1, 8.8.15775096184361.1, 8.0.194754273881.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
41Data not computed