Properties

 Label 16.0.28624230515...5793.1 Degree $16$ Signature $[0, 8]$ Discriminant $17^{15}$ Root discriminant $14.24$ Ramified prime $17$ Class number $1$ Class group Trivial Galois group $C_{16}$ (as 16T1)

Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

gp: K = bnfinit(x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]);

Normalizeddefining polynomial

$$x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2862423051509815793=17^{15}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $14.24$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $17$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $16$ This field is Galois and abelian over $\Q$. Conductor: $$17$$ Dirichlet character group: $\lbrace$$\chi_{17}(1,·), \chi_{17}(2,·), \chi_{17}(3,·), \chi_{17}(4,·), \chi_{17}(5,·), \chi_{17}(6,·), \chi_{17}(7,·), \chi_{17}(8,·), \chi_{17}(9,·), \chi_{17}(10,·), \chi_{17}(11,·), \chi_{17}(12,·), \chi_{17}(13,·), \chi_{17}(14,·), \chi_{17}(15,·), \chi_{17}(16,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $7$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$a$$ (order $34$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{9} - a^{8}$$,  $$a^{10} + a^{8}$$,  $$a^{15} + a^{7}$$,  $$a^{12} - a^{7} - a$$,  $$a^{11} - a^{10} + a^{9}$$,  $$a^{14} - a^{13} + a^{12} - a^{11} + a^{10} - a^{9} + a^{8}$$,  $$a^{14} - a^{9} + a^{2}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3640.01221338$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 16 The 16 conjugacy class representatives for $C_{16}$ Character table for $C_{16}$

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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.

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]])

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];

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed