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Properties

Label 16.0.66776644863...1872.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}\cdot 3^{8}\cdot 17^{14}$
Root discriminant $633.20$
Ramified primes $2, 3, 17$
Class number $1520548928$ (GRH)
Class group $[2, 2, 2, 190068616]$ (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![211215000911202, 0, 63887310947520, 0, 6070515141456, 0, 233097458400, 0, 4468073076, 0, 46693152, 0, 270504, 0, 816, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 816*x^14 + 270504*x^12 + 46693152*x^10 + 4468073076*x^8 + 233097458400*x^6 + 6070515141456*x^4 + 63887310947520*x^2 + 211215000911202)
 
gp: K = bnfinit(x^16 + 816*x^14 + 270504*x^12 + 46693152*x^10 + 4468073076*x^8 + 233097458400*x^6 + 6070515141456*x^4 + 63887310947520*x^2 + 211215000911202, 1)
 

Normalized defining polynomial

\( x^{16} + 816 x^{14} + 270504 x^{12} + 46693152 x^{10} + 4468073076 x^{8} + 233097458400 x^{6} + 6070515141456 x^{4} + 63887310947520 x^{2} + 211215000911202 \)

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:  \(667766448639661081824497937039393690193231872=2^{79}\cdot 3^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $633.20$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3264=2^{6}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3264}(1,·)$, $\chi_{3264}(2117,·)$, $\chi_{3264}(1993,·)$, $\chi_{3264}(461,·)$, $\chi_{3264}(3025,·)$, $\chi_{3264}(3221,·)$, $\chi_{3264}(217,·)$, $\chi_{3264}(797,·)$, $\chi_{3264}(1633,·)$, $\chi_{3264}(485,·)$, $\chi_{3264}(361,·)$, $\chi_{3264}(2093,·)$, $\chi_{3264}(1393,·)$, $\chi_{3264}(1589,·)$, $\chi_{3264}(1849,·)$, $\chi_{3264}(2429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{63} a^{4} - \frac{2}{21} a^{2} - \frac{3}{7}$, $\frac{1}{63} a^{5} - \frac{2}{21} a^{3} - \frac{3}{7} a$, $\frac{1}{189} a^{6} + \frac{1}{7}$, $\frac{1}{189} a^{7} + \frac{1}{7} a$, $\frac{1}{2766393} a^{8} + \frac{8}{54243} a^{6} + \frac{53}{18081} a^{4} - \frac{829}{6027} a^{2} - \frac{263}{2009}$, $\frac{1}{2766393} a^{9} + \frac{8}{54243} a^{7} + \frac{53}{18081} a^{5} - \frac{829}{6027} a^{3} - \frac{263}{2009} a$, $\frac{1}{8299179} a^{10} + \frac{113}{54243} a^{6} - \frac{1}{18081} a^{4} - \frac{23}{6027} a^{2} + \frac{754}{2009}$, $\frac{1}{141086043} a^{11} - \frac{44}{54243} a^{7} + \frac{15}{2009} a^{5} - \frac{199}{34153} a^{3} - \frac{766}{2009} a$, $\frac{1}{20739648321} a^{12} + \frac{4}{135553257} a^{10} - \frac{13}{135553257} a^{8} - \frac{88}{42189} a^{6} + \frac{12527}{15061473} a^{4} + \frac{8832}{98441} a^{2} + \frac{5737}{98441}$, $\frac{1}{9104705612919} a^{13} + \frac{151}{59507879823} a^{11} - \frac{2659}{59507879823} a^{9} - \frac{5660}{18520971} a^{7} - \frac{21743767}{6611986647} a^{5} - \frac{16578281}{129646797} a^{3} + \frac{8206279}{43215599} a$, $\frac{1}{464339986258869} a^{14} + \frac{2}{1300672230417} a^{12} - \frac{121}{59507879823} a^{10} - \frac{6697}{59507879823} a^{8} + \frac{593132803}{337211318997} a^{6} + \frac{16297990}{6611986647} a^{4} + \frac{2458160}{43215599} a^{2} - \frac{38616}{98441}$, $\frac{1}{464339986258869} a^{15} - \frac{6430}{3034901870973} a^{11} + \frac{334}{2203995549} a^{9} + \frac{618452671}{337211318997} a^{7} + \frac{7802399}{2203995549} a^{5} + \frac{216806635}{2203995549} a^{3} + \frac{13681585}{43215599} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{190068616}$, which has order $1520548928$ (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:  \( 55546734.0870713 \) (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

\(\Q(\sqrt{2}) \), 4.4.591872.2, 8.8.51835034729971712.3

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 R R $16$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ $16$ $16$ R $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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
2Data not computed
3Data not computed
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$