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Properties

Label 16.16.6332511891...4593.1
Degree $16$
Signature $[16, 0]$
Discriminant $97^{15}$
Root discriminant $72.88$
Ramified prime $97$
Class number $1$ (GRH)
Class group Trivial (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![3721, -17690, 15475, 43324, -86644, 18576, 63779, -44003, -9748, 17205, -2576, -2183, 650, 98, -45, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 45*x^14 + 98*x^13 + 650*x^12 - 2183*x^11 - 2576*x^10 + 17205*x^9 - 9748*x^8 - 44003*x^7 + 63779*x^6 + 18576*x^5 - 86644*x^4 + 43324*x^3 + 15475*x^2 - 17690*x + 3721)
 
gp: K = bnfinit(x^16 - x^15 - 45*x^14 + 98*x^13 + 650*x^12 - 2183*x^11 - 2576*x^10 + 17205*x^9 - 9748*x^8 - 44003*x^7 + 63779*x^6 + 18576*x^5 - 86644*x^4 + 43324*x^3 + 15475*x^2 - 17690*x + 3721, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 45 x^{14} + 98 x^{13} + 650 x^{12} - 2183 x^{11} - 2576 x^{10} + 17205 x^{9} - 9748 x^{8} - 44003 x^{7} + 63779 x^{6} + 18576 x^{5} - 86644 x^{4} + 43324 x^{3} + 15475 x^{2} - 17690 x + 3721 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(633251189136789386043275954593=97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.88$
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:  $97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(97\)
Dirichlet character group:    $\lbrace$$\chi_{97}(64,·)$, $\chi_{97}(1,·)$, $\chi_{97}(70,·)$, $\chi_{97}(8,·)$, $\chi_{97}(75,·)$, $\chi_{97}(12,·)$, $\chi_{97}(79,·)$, $\chi_{97}(18,·)$, $\chi_{97}(85,·)$, $\chi_{97}(22,·)$, $\chi_{97}(89,·)$, $\chi_{97}(27,·)$, $\chi_{97}(96,·)$, $\chi_{97}(33,·)$, $\chi_{97}(47,·)$, $\chi_{97}(50,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{18712565507118089} a^{15} + \frac{4702651407010647}{18712565507118089} a^{14} - \frac{1858332681867168}{18712565507118089} a^{13} + \frac{2136683603725307}{18712565507118089} a^{12} - \frac{1953063510001550}{18712565507118089} a^{11} - \frac{6495986396811465}{18712565507118089} a^{10} - \frac{60411879064726}{18712565507118089} a^{9} + \frac{2763854199722048}{18712565507118089} a^{8} + \frac{8672647571520568}{18712565507118089} a^{7} + \frac{5349577462317796}{18712565507118089} a^{6} - \frac{5505523584961902}{18712565507118089} a^{5} + \frac{1018921141280993}{18712565507118089} a^{4} + \frac{3204798404152637}{18712565507118089} a^{3} - \frac{529730990245149}{18712565507118089} a^{2} + \frac{9323578277179799}{18712565507118089} a + \frac{133891734766424}{306763368969149}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 4029722860.88 \) (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

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
97Data not computed