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Properties

Label 16.0.23349664196...6753.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 17^{15}$
Root discriminant $51.35$
Ramified primes $13, 17$
Class number $1808$ (GRH)
Class group $[2, 2, 2, 226]$ (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![1439629, -1328092, 1328092, -881944, 881944, -361438, 361438, -88792, 88792, -13057, 13057, -1123, 1123, -52, 52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629)
 
gp: K = bnfinit(x^16 - x^15 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 52 x^{14} - 52 x^{13} + 1123 x^{12} - 1123 x^{11} + 13057 x^{10} - 13057 x^{9} + 88792 x^{8} - 88792 x^{7} + 361438 x^{6} - 361438 x^{5} + 881944 x^{4} - 881944 x^{3} + 1328092 x^{2} - 1328092 x + 1439629 \)

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:  \(2334966419615122175401076753=13^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.35$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(66,·)$, $\chi_{221}(196,·)$, $\chi_{221}(129,·)$, $\chi_{221}(12,·)$, $\chi_{221}(194,·)$, $\chi_{221}(142,·)$, $\chi_{221}(207,·)$, $\chi_{221}(144,·)$, $\chi_{221}(90,·)$, $\chi_{221}(157,·)$, $\chi_{221}(116,·)$, $\chi_{221}(53,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$, $\chi_{221}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{399331} a^{9} + \frac{80968}{399331} a^{8} + \frac{27}{399331} a^{7} - \frac{53423}{399331} a^{6} + \frac{243}{399331} a^{5} + \frac{198324}{399331} a^{4} + \frac{810}{399331} a^{3} - \frac{162952}{399331} a^{2} + \frac{729}{399331} a - \frac{61107}{399331}$, $\frac{1}{399331} a^{10} + \frac{30}{399331} a^{8} + \frac{156427}{399331} a^{7} + \frac{315}{399331} a^{6} + \frac{90319}{399331} a^{5} + \frac{1350}{399331} a^{4} + \frac{142583}{399331} a^{3} + \frac{2025}{399331} a^{2} + \frac{14209}{399331} a + \frac{486}{399331}$, $\frac{1}{399331} a^{11} + \frac{123373}{399331} a^{8} - \frac{495}{399331} a^{7} + \frac{95685}{399331} a^{6} - \frac{5940}{399331} a^{5} + \frac{182828}{399331} a^{4} - \frac{22275}{399331} a^{3} + \frac{110797}{399331} a^{2} - \frac{21384}{399331} a - \frac{163445}{399331}$, $\frac{1}{399331} a^{12} - \frac{594}{399331} a^{8} - \frac{40738}{399331} a^{7} - \frac{8316}{399331} a^{6} + \frac{153014}{399331} a^{5} - \frac{40095}{399331} a^{4} + \frac{11417}{399331} a^{3} - \frac{64152}{399331} a^{2} + \frac{146444}{399331} a - \frac{16038}{399331}$, $\frac{1}{399331} a^{13} + \frac{134534}{399331} a^{8} + \frac{7722}{399331} a^{7} - \frac{33099}{399331} a^{6} + \frac{104247}{399331} a^{5} + \frac{13228}{399331} a^{4} + \frac{17657}{399331} a^{3} - \frac{8942}{399331} a^{2} + \frac{17657}{399331} a + \frac{41563}{399331}$, $\frac{1}{399331} a^{14} + \frac{9828}{399331} a^{8} - \frac{71538}{399331} a^{7} + \frac{154791}{399331} a^{6} + \frac{66608}{399331} a^{5} - \frac{2594}{399331} a^{4} + \frac{35881}{399331} a^{3} + \frac{128787}{399331} a^{2} - \frac{197628}{399331} a - \frac{58159}{399331}$, $\frac{1}{399331} a^{15} + \frac{41641}{399331} a^{8} - \frac{110565}{399331} a^{7} - \frac{12413}{399331} a^{6} + \frac{5188}{399331} a^{5} + \frac{42220}{399331} a^{4} + \frac{154727}{399331} a^{3} - \frac{22682}{399331} a^{2} - \frac{34813}{399331} a - \frac{34228}{399331}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{226}$, which has order $1808$ (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:  \( 3640.01221338 \) (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, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R 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.

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
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17Data not computed