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Properties

Label 16.0.22434050585...5073.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 97^{8}$
Root discriminant $140.26$
Ramified primes $17, 97$
Class number $5826818$ (GRH)
Class group $[17, 17, 20162]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2952618490201, -1081338149209, 1081338149209, -145697978713, 145697978713, -9250453849, 9250453849, -316389721, 316389721, -6179161, 6179161, -68953, 68953, -409, 409, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 409*x^14 - 409*x^13 + 68953*x^12 - 68953*x^11 + 6179161*x^10 - 6179161*x^9 + 316389721*x^8 - 316389721*x^7 + 9250453849*x^6 - 9250453849*x^5 + 145697978713*x^4 - 145697978713*x^3 + 1081338149209*x^2 - 1081338149209*x + 2952618490201)
 
gp: K = bnfinit(x^16 - x^15 + 409*x^14 - 409*x^13 + 68953*x^12 - 68953*x^11 + 6179161*x^10 - 6179161*x^9 + 316389721*x^8 - 316389721*x^7 + 9250453849*x^6 - 9250453849*x^5 + 145697978713*x^4 - 145697978713*x^3 + 1081338149209*x^2 - 1081338149209*x + 2952618490201, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 409 x^{14} - 409 x^{13} + 68953 x^{12} - 68953 x^{11} + 6179161 x^{10} - 6179161 x^{9} + 316389721 x^{8} - 316389721 x^{7} + 9250453849 x^{6} - 9250453849 x^{5} + 145697978713 x^{4} - 145697978713 x^{3} + 1081338149209 x^{2} - 1081338149209 x + 2952618490201 \)

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:  \(22434050585222044572730141713145073=17^{15}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.26$
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:  $17, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1649=17\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{1649}(1,·)$, $\chi_{1649}(195,·)$, $\chi_{1649}(389,·)$, $\chi_{1649}(775,·)$, $\chi_{1649}(1163,·)$, $\chi_{1649}(1165,·)$, $\chi_{1649}(1357,·)$, $\chi_{1649}(193,·)$, $\chi_{1649}(1262,·)$, $\chi_{1649}(1359,·)$, $\chi_{1649}(581,·)$, $\chi_{1649}(96,·)$, $\chi_{1649}(98,·)$, $\chi_{1649}(872,·)$, $\chi_{1649}(1066,·)$, $\chi_{1649}(971,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{319166878249} a^{9} - \frac{136629353176}{319166878249} a^{8} + \frac{216}{319166878249} a^{7} - \frac{61151793374}{319166878249} a^{6} + \frac{15552}{319166878249} a^{5} - \frac{158271941701}{319166878249} a^{4} + \frac{414720}{319166878249} a^{3} + \frac{89014126181}{319166878249} a^{2} + \frac{2985984}{319166878249} a - \frac{52124499706}{319166878249}$, $\frac{1}{319166878249} a^{10} + \frac{240}{319166878249} a^{8} + \frac{87435693734}{319166878249} a^{7} + \frac{20160}{319166878249} a^{6} + \frac{7520147858}{319166878249} a^{5} + \frac{691200}{319166878249} a^{4} + \frac{41800218935}{319166878249} a^{3} + \frac{8294400}{319166878249} a^{2} - \frac{136731129278}{319166878249} a + \frac{15925248}{319166878249}$, $\frac{1}{319166878249} a^{11} + \frac{4291996327}{319166878249} a^{8} - \frac{31680}{319166878249} a^{7} + \frac{2274158164}{319166878249} a^{6} - \frac{3041280}{319166878249} a^{5} + \frac{46207715544}{319166878249} a^{4} - \frac{91238400}{319166878249} a^{3} - \frac{115940570035}{319166878249} a^{2} - \frac{700710912}{319166878249} a + \frac{62371677729}{319166878249}$, $\frac{1}{319166878249} a^{12} - \frac{38016}{319166878249} a^{8} + \frac{32703586279}{319166878249} a^{7} - \frac{4257792}{319166878249} a^{6} + \frac{2958392081}{319166878249} a^{5} - \frac{164229120}{319166878249} a^{4} - \frac{98977308802}{319166878249} a^{3} - \frac{2102132736}{319166878249} a^{2} + \frac{56840407307}{319166878249} a - \frac{4204265472}{319166878249}$, $\frac{1}{319166878249} a^{13} + \frac{52989871689}{319166878249} a^{8} + \frac{3953664}{319166878249} a^{7} + \frac{67922651813}{319166878249} a^{6} + \frac{426995712}{319166878249} a^{5} - \frac{31124263870}{319166878249} a^{4} + \frac{13663862784}{319166878249} a^{3} - \frac{108548769944}{319166878249} a^{2} + \frac{109310902272}{319166878249} a + \frac{142166224745}{319166878249}$, $\frac{1}{319166878249} a^{14} + \frac{5031936}{319166878249} a^{8} + \frac{112117983953}{319166878249} a^{7} + \frac{634023936}{319166878249} a^{6} - \frac{40729132280}{319166878249} a^{5} + \frac{26085556224}{319166878249} a^{4} - \frac{151900675378}{319166878249} a^{3} + \frac{28640538071}{319166878249} a^{2} - \frac{106134118730}{319166878249} a + \frac{77155785646}{319166878249}$, $\frac{1}{319166878249} a^{15} - \frac{77743836733}{319166878249} a^{8} - \frac{452874240}{319166878249} a^{7} - \frac{109174583606}{319166878249} a^{6} - \frac{52171112448}{319166878249} a^{5} - \frac{109458337701}{319166878249} a^{4} - \frac{143202690355}{319166878249} a^{3} - \frac{121075118779}{319166878249} a^{2} + \frac{52718678325}{319166878249} a - \frac{44823000147}{319166878249}$

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

Class group and class number

$C_{17}\times C_{17}\times C_{20162}$, which has order $5826818$ (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:  \( -1 \) (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
97Data not computed