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Properties

Label 16.16.6988337528...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 17^{15}$
Root discriminant $47.62$
Ramified primes $5, 17$
Class number $2$ (GRH)
Class group $[2]$ (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![1, -69, 732, -2398, 1412, 5235, -5677, -4506, 5305, 1886, -2107, -375, 392, 33, -33, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 33*x^14 + 33*x^13 + 392*x^12 - 375*x^11 - 2107*x^10 + 1886*x^9 + 5305*x^8 - 4506*x^7 - 5677*x^6 + 5235*x^5 + 1412*x^4 - 2398*x^3 + 732*x^2 - 69*x + 1)
 
gp: K = bnfinit(x^16 - x^15 - 33*x^14 + 33*x^13 + 392*x^12 - 375*x^11 - 2107*x^10 + 1886*x^9 + 5305*x^8 - 4506*x^7 - 5677*x^6 + 5235*x^5 + 1412*x^4 - 2398*x^3 + 732*x^2 - 69*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 33 x^{14} + 33 x^{13} + 392 x^{12} - 375 x^{11} - 2107 x^{10} + 1886 x^{9} + 5305 x^{8} - 4506 x^{7} - 5677 x^{6} + 5235 x^{5} + 1412 x^{4} - 2398 x^{3} + 732 x^{2} - 69 x + 1 \)

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:  \(698833752810013621337890625=5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.62$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(9,·)$, $\chi_{85}(12,·)$, $\chi_{85}(78,·)$, $\chi_{85}(16,·)$, $\chi_{85}(81,·)$, $\chi_{85}(82,·)$, $\chi_{85}(19,·)$, $\chi_{85}(21,·)$, $\chi_{85}(22,·)$, $\chi_{85}(23,·)$, $\chi_{85}(28,·)$, $\chi_{85}(37,·)$, $\chi_{85}(49,·)$, $\chi_{85}(58,·)$, $\chi_{85}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} + \frac{3}{13} a^{9} - \frac{2}{13} a^{8} - \frac{4}{13} a^{6} + \frac{1}{13} a^{5} - \frac{5}{13} a^{4} + \frac{3}{13} a^{2} - \frac{4}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{11} + \frac{2}{13} a^{9} + \frac{6}{13} a^{8} - \frac{4}{13} a^{7} + \frac{5}{13} a^{5} + \frac{2}{13} a^{4} + \frac{3}{13} a^{3} + \frac{6}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{12} - \frac{1}{13}$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{2041} a^{14} + \frac{46}{2041} a^{13} + \frac{60}{2041} a^{12} - \frac{5}{2041} a^{11} + \frac{47}{2041} a^{10} + \frac{404}{2041} a^{9} - \frac{449}{2041} a^{8} - \frac{71}{2041} a^{7} + \frac{410}{2041} a^{6} + \frac{581}{2041} a^{5} + \frac{730}{2041} a^{4} - \frac{587}{2041} a^{3} - \frac{302}{2041} a^{2} - \frac{784}{2041} a - \frac{185}{2041}$, $\frac{1}{645362159} a^{15} - \frac{142749}{645362159} a^{14} + \frac{49097}{49643243} a^{13} - \frac{19873275}{645362159} a^{12} - \frac{18473262}{645362159} a^{11} - \frac{13319405}{645362159} a^{10} + \frac{302209321}{645362159} a^{9} + \frac{188418221}{645362159} a^{8} - \frac{215294305}{645362159} a^{7} - \frac{130675565}{645362159} a^{6} - \frac{171421047}{645362159} a^{5} - \frac{207494342}{645362159} a^{4} + \frac{268896673}{645362159} a^{3} + \frac{249085640}{645362159} a^{2} - \frac{124837875}{645362159} a - \frac{1439121}{645362159}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 165867618.848 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ 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
5Data not computed
17Data not computed