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Properties

Label 16.0.14360793917...2689.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 31^{8}$
Root discriminant $66.42$
Ramified primes $17, 31$
Class number $20655$ (GRH)
Class group $[3, 3, 3, 765]$ (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![72060551, -29325087, 47580730, -17463475, 14901406, -4845620, 2860699, -810784, 366078, -88467, 31872, -6330, 1839, -278, 64, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 64*x^14 - 278*x^13 + 1839*x^12 - 6330*x^11 + 31872*x^10 - 88467*x^9 + 366078*x^8 - 810784*x^7 + 2860699*x^6 - 4845620*x^5 + 14901406*x^4 - 17463475*x^3 + 47580730*x^2 - 29325087*x + 72060551)
 
gp: K = bnfinit(x^16 - 6*x^15 + 64*x^14 - 278*x^13 + 1839*x^12 - 6330*x^11 + 31872*x^10 - 88467*x^9 + 366078*x^8 - 810784*x^7 + 2860699*x^6 - 4845620*x^5 + 14901406*x^4 - 17463475*x^3 + 47580730*x^2 - 29325087*x + 72060551, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 64 x^{14} - 278 x^{13} + 1839 x^{12} - 6330 x^{11} + 31872 x^{10} - 88467 x^{9} + 366078 x^{8} - 810784 x^{7} + 2860699 x^{6} - 4845620 x^{5} + 14901406 x^{4} - 17463475 x^{3} + 47580730 x^{2} - 29325087 x + 72060551 \)

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:  \(143607939176308221946269182689=17^{14}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.42$
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:  $17, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(527=17\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{527}(1,·)$, $\chi_{527}(526,·)$, $\chi_{527}(404,·)$, $\chi_{527}(342,·)$, $\chi_{527}(280,·)$, $\chi_{527}(154,·)$, $\chi_{527}(94,·)$, $\chi_{527}(32,·)$, $\chi_{527}(433,·)$, $\chi_{527}(495,·)$, $\chi_{527}(497,·)$, $\chi_{527}(30,·)$, $\chi_{527}(247,·)$, $\chi_{527}(185,·)$, $\chi_{527}(123,·)$, $\chi_{527}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2166221443622873063273220170577147797497618} a^{15} + \frac{466177913608339633037384392312956902713569}{2166221443622873063273220170577147797497618} a^{14} + \frac{74324906138316634280816170117444619846513}{1083110721811436531636610085288573898748809} a^{13} - \frac{219875024824497952378956928336464226135344}{1083110721811436531636610085288573898748809} a^{12} + \frac{1001408752610611086047882867395260497992093}{2166221443622873063273220170577147797497618} a^{11} - \frac{311690817674710909517414180203573892250215}{1083110721811436531636610085288573898748809} a^{10} - \frac{284688731728291670383873831728029167522360}{1083110721811436531636610085288573898748809} a^{9} + \frac{182740776414865221889112438325742168710631}{2166221443622873063273220170577147797497618} a^{8} - \frac{271356127594295356677547543688920941991504}{1083110721811436531636610085288573898748809} a^{7} - \frac{333138439032500037636646671623782632298329}{1083110721811436531636610085288573898748809} a^{6} - \frac{199554103609786074555316640973351790251693}{2166221443622873063273220170577147797497618} a^{5} + \frac{270719701042657602675571279342348579556697}{1083110721811436531636610085288573898748809} a^{4} - \frac{164778901610915162386599892227201921962318}{1083110721811436531636610085288573898748809} a^{3} + \frac{950067427945065388112497362648303456007757}{2166221443622873063273220170577147797497618} a^{2} - \frac{329999923677030971218563226197952357409683}{1083110721811436531636610085288573898748809} a - \frac{85278726104847795305553117012668489388569}{2166221443622873063273220170577147797497618}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{765}$, which has order $20655$ (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

$C_2\times C_8$ (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

\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-527}) \), \(\Q(\sqrt{17}, \sqrt{-31})\), 4.4.4913.1, 4.0.4721393.1, 8.0.22291551860449.1, \(\Q(\zeta_{17})^+\), 8.0.378956381627633.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
31Data not computed