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Properties

Label 16.0.45850482521...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 17^{15}$
Root discriminant $82.48$
Ramified primes $3, 5, 17$
Class number $2824$ (GRH)
Class group $[2, 1412]$ (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![2213401, -1949034, 3292187, -771733, 1553597, -463965, 540278, -83556, 86310, -13839, 11748, -1395, 1072, -52, 52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1072*x^12 - 1395*x^11 + 11748*x^10 - 13839*x^9 + 86310*x^8 - 83556*x^7 + 540278*x^6 - 463965*x^5 + 1553597*x^4 - 771733*x^3 + 3292187*x^2 - 1949034*x + 2213401)
 
gp: K = bnfinit(x^16 - x^15 + 52*x^14 - 52*x^13 + 1072*x^12 - 1395*x^11 + 11748*x^10 - 13839*x^9 + 86310*x^8 - 83556*x^7 + 540278*x^6 - 463965*x^5 + 1553597*x^4 - 771733*x^3 + 3292187*x^2 - 1949034*x + 2213401, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 52 x^{14} - 52 x^{13} + 1072 x^{12} - 1395 x^{11} + 11748 x^{10} - 13839 x^{9} + 86310 x^{8} - 83556 x^{7} + 540278 x^{6} - 463965 x^{5} + 1553597 x^{4} - 771733 x^{3} + 3292187 x^{2} - 1949034 x + 2213401 \)

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:  \(4585048252186499369597900390625=3^{8}\cdot 5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.48$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(197,·)$, $\chi_{255}(16,·)$, $\chi_{255}(19,·)$, $\chi_{255}(218,·)$, $\chi_{255}(92,·)$, $\chi_{255}(94,·)$, $\chi_{255}(227,·)$, $\chi_{255}(229,·)$, $\chi_{255}(166,·)$, $\chi_{255}(233,·)$, $\chi_{255}(106,·)$, $\chi_{255}(173,·)$, $\chi_{255}(49,·)$, $\chi_{255}(158,·)$, $\chi_{255}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{289034445079918768974533783036378799974459625164723} a^{15} - \frac{7805059681154855318417728924119045946318528872880}{289034445079918768974533783036378799974459625164723} a^{14} + \frac{139421278639331645414297196489737606004470479679180}{289034445079918768974533783036378799974459625164723} a^{13} + \frac{37135064956533850334366520060328602647135011429108}{289034445079918768974533783036378799974459625164723} a^{12} - \frac{127317722405057512291145412440779026779535404574539}{289034445079918768974533783036378799974459625164723} a^{11} + \frac{15506641334413197413843062501780127517421217998640}{289034445079918768974533783036378799974459625164723} a^{10} + \frac{83252585739488419017947772572439068244862566449089}{289034445079918768974533783036378799974459625164723} a^{9} - \frac{96795861642757680756201663079753241428964024245772}{289034445079918768974533783036378799974459625164723} a^{8} - \frac{129484990356321113194159276588720023209883618025216}{289034445079918768974533783036378799974459625164723} a^{7} - \frac{133738533940126421812790530054608978851819731939013}{289034445079918768974533783036378799974459625164723} a^{6} + \frac{90871878891179884722960847541927411593620119148138}{289034445079918768974533783036378799974459625164723} a^{5} - \frac{80685588460940467254233631493707288531404436488694}{289034445079918768974533783036378799974459625164723} a^{4} - \frac{29179282860179436161656724368953597311278075696658}{289034445079918768974533783036378799974459625164723} a^{3} + \frac{96835165990098560863202421181736244314785044444869}{289034445079918768974533783036378799974459625164723} a^{2} - \frac{132729208814143006860330249003391962149330256830075}{289034445079918768974533783036378799974459625164723} a - \frac{135826115220380632656635164545256289922518634717000}{289034445079918768974533783036378799974459625164723}$

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

Class group and class number

$C_{2}\times C_{1412}$, which has order $2824$ (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:  \( 81485.0410293661 \) (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, 8.8.256461670625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
5Data not computed
17Data not computed