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Properties

Label 16.16.6135858022...7633.1
Degree $16$
Signature $[16, 0]$
Discriminant $11^{8}\cdot 17^{15}$
Root discriminant $47.23$
Ramified primes $11, 17$
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![-21929, 133466, -133466, -312682, 312682, 207824, -207824, -64822, 64822, 10913, -10913, -1021, 1021, 50, -50, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 50*x^14 + 50*x^13 + 1021*x^12 - 1021*x^11 - 10913*x^10 + 10913*x^9 + 64822*x^8 - 64822*x^7 - 207824*x^6 + 207824*x^5 + 312682*x^4 - 312682*x^3 - 133466*x^2 + 133466*x - 21929)
 
gp: K = bnfinit(x^16 - x^15 - 50*x^14 + 50*x^13 + 1021*x^12 - 1021*x^11 - 10913*x^10 + 10913*x^9 + 64822*x^8 - 64822*x^7 - 207824*x^6 + 207824*x^5 + 312682*x^4 - 312682*x^3 - 133466*x^2 + 133466*x - 21929, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 50 x^{14} + 50 x^{13} + 1021 x^{12} - 1021 x^{11} - 10913 x^{10} + 10913 x^{9} + 64822 x^{8} - 64822 x^{7} - 207824 x^{6} + 207824 x^{5} + 312682 x^{4} - 312682 x^{3} - 133466 x^{2} + 133466 x - 21929 \)

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:  \(613585802270249473903607633=11^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.23$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(65,·)$, $\chi_{187}(10,·)$, $\chi_{187}(142,·)$, $\chi_{187}(144,·)$, $\chi_{187}(131,·)$, $\chi_{187}(89,·)$, $\chi_{187}(100,·)$, $\chi_{187}(111,·)$, $\chi_{187}(155,·)$, $\chi_{187}(164,·)$, $\chi_{187}(166,·)$, $\chi_{187}(109,·)$, $\chi_{187}(175,·)$, $\chi_{187}(54,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1801} a^{9} + \frac{461}{1801} a^{8} - \frac{27}{1801} a^{7} - \frac{258}{1801} a^{6} + \frac{243}{1801} a^{5} + \frac{134}{1801} a^{4} - \frac{810}{1801} a^{3} + \frac{759}{1801} a^{2} + \frac{729}{1801} a + \frac{841}{1801}$, $\frac{1}{1801} a^{10} - \frac{30}{1801} a^{8} - \frac{418}{1801} a^{7} + \frac{315}{1801} a^{6} - \frac{227}{1801} a^{5} + \frac{451}{1801} a^{4} - \frac{439}{1801} a^{3} + \frac{224}{1801} a^{2} - \frac{242}{1801} a - \frac{486}{1801}$, $\frac{1}{1801} a^{11} + \frac{805}{1801} a^{8} - \frac{495}{1801} a^{7} - \frac{763}{1801} a^{6} + \frac{537}{1801} a^{5} - \frac{21}{1801} a^{4} - \frac{663}{1801} a^{3} - \frac{885}{1801} a^{2} - \frac{228}{1801} a + \frac{16}{1801}$, $\frac{1}{1801} a^{12} - \frac{594}{1801} a^{8} - \frac{640}{1801} a^{7} - \frac{689}{1801} a^{6} + \frac{673}{1801} a^{5} - \frac{473}{1801} a^{4} - \frac{797}{1801} a^{3} - \frac{684}{1801} a^{2} + \frac{297}{1801} a + \frac{171}{1801}$, $\frac{1}{1801} a^{13} - \frac{558}{1801} a^{8} - \frac{518}{1801} a^{7} + \frac{506}{1801} a^{6} - \frac{211}{1801} a^{5} - \frac{445}{1801} a^{4} + \frac{844}{1801} a^{3} + \frac{893}{1801} a^{2} - \frac{844}{1801} a + \frac{677}{1801}$, $\frac{1}{1801} a^{14} - \frac{823}{1801} a^{8} - \frac{152}{1801} a^{7} - \frac{95}{1801} a^{6} + \frac{74}{1801} a^{5} - \frac{26}{1801} a^{4} - \frac{837}{1801} a^{3} - \frac{557}{1801} a^{2} + \frac{433}{1801} a - \frac{783}{1801}$, $\frac{1}{1801} a^{15} - \frac{760}{1801} a^{8} - \frac{704}{1801} a^{7} + \frac{258}{1801} a^{6} + \frac{52}{1801} a^{5} - \frac{416}{1801} a^{4} - \frac{817}{1801} a^{3} + \frac{143}{1801} a^{2} - \frac{549}{1801} a + \frac{559}{1801}$

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:  \( 124319056.419 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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.

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
11Data not computed
17Data not computed