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Properties

Label 16.16.2441334965...5713.1
Degree $16$
Signature $[16, 0]$
Discriminant $17^{15}\cdot 31^{8}$
Root discriminant $79.29$
Ramified primes $17, 31$
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![11409721, 273802951, -273802951, -154016057, 154016057, 33154759, -33154759, -3610937, 3610937, 218823, -218823, -7481, 7481, 135, -135, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 135*x^14 + 135*x^13 + 7481*x^12 - 7481*x^11 - 218823*x^10 + 218823*x^9 + 3610937*x^8 - 3610937*x^7 - 33154759*x^6 + 33154759*x^5 + 154016057*x^4 - 154016057*x^3 - 273802951*x^2 + 273802951*x + 11409721)
 
gp: K = bnfinit(x^16 - x^15 - 135*x^14 + 135*x^13 + 7481*x^12 - 7481*x^11 - 218823*x^10 + 218823*x^9 + 3610937*x^8 - 3610937*x^7 - 33154759*x^6 + 33154759*x^5 + 154016057*x^4 - 154016057*x^3 - 273802951*x^2 + 273802951*x + 11409721, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 135 x^{14} + 135 x^{13} + 7481 x^{12} - 7481 x^{11} - 218823 x^{10} + 218823 x^{9} + 3610937 x^{8} - 3610937 x^{7} - 33154759 x^{6} + 33154759 x^{5} + 154016057 x^{4} - 154016057 x^{3} - 273802951 x^{2} + 273802951 x + 11409721 \)

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:  \(2441334965997239773086576105713=17^{15}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.29$
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:  $17, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(527=17\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{527}(1,·)$, $\chi_{527}(278,·)$, $\chi_{527}(464,·)$, $\chi_{527}(280,·)$, $\chi_{527}(402,·)$, $\chi_{527}(404,·)$, $\chi_{527}(342,·)$, $\chi_{527}(216,·)$, $\chi_{527}(92,·)$, $\chi_{527}(94,·)$, $\chi_{527}(32,·)$, $\chi_{527}(497,·)$, $\chi_{527}(371,·)$, $\chi_{527}(309,·)$, $\chi_{527}(61,·)$, $\chi_{527}(373,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{16922039} a^{9} - \frac{3128542}{16922039} a^{8} - \frac{72}{16922039} a^{7} - \frac{2837780}{16922039} a^{6} + \frac{1728}{16922039} a^{5} + \frac{5989483}{16922039} a^{4} - \frac{15360}{16922039} a^{3} - \frac{7873021}{16922039} a^{2} + \frac{36864}{16922039} a + \frac{7873021}{16922039}$, $\frac{1}{16922039} a^{10} - \frac{80}{16922039} a^{8} - \frac{8106297}{16922039} a^{7} + \frac{2240}{16922039} a^{6} - \frac{2942421}{16922039} a^{5} - \frac{25600}{16922039} a^{4} - \frac{3687381}{16922039} a^{3} + \frac{102400}{16922039} a^{2} - \frac{2172515}{16922039} a - \frac{65536}{16922039}$, $\frac{1}{16922039} a^{11} - \frac{4559072}{16922039} a^{8} - \frac{3520}{16922039} a^{7} + \frac{6943725}{16922039} a^{6} + \frac{112640}{16922039} a^{5} + \frac{1654167}{16922039} a^{4} - \frac{1126400}{16922039} a^{3} - \frac{5898752}{16922039} a^{2} + \frac{2883584}{16922039} a + \frac{3726237}{16922039}$, $\frac{1}{16922039} a^{12} - \frac{4224}{16922039} a^{8} + \frac{209282}{16922039} a^{7} + \frac{157696}{16922039} a^{6} - \frac{5939591}{16922039} a^{5} - \frac{2027520}{16922039} a^{4} + \frac{7074749}{16922039} a^{3} - \frac{8271287}{16922039} a^{2} - \frac{334903}{16922039} a - \frac{5767168}{16922039}$, $\frac{1}{16922039} a^{13} + \frac{1360333}{16922039} a^{8} - \frac{146432}{16922039} a^{7} + \frac{5003340}{16922039} a^{6} + \frac{5271552}{16922039} a^{5} + \frac{8202636}{16922039} a^{4} - \frac{5463771}{16922039} a^{3} - \frac{4168972}{16922039} a^{2} - \frac{2351983}{16922039} a + \frac{3834069}{16922039}$, $\frac{1}{16922039} a^{14} - \frac{186368}{16922039} a^{8} + \frac{1415082}{16922039} a^{7} + \frac{7827456}{16922039} a^{6} - \frac{7211406}{16922039} a^{5} - \frac{5815734}{16922039} a^{4} - \frac{8172257}{16922039} a^{3} + \frac{3284988}{16922039} a^{2} - \frac{3480086}{16922039} a - \frac{5636971}{16922039}$, $\frac{1}{16922039} a^{15} + \frac{7075410}{16922039} a^{8} - \frac{5591040}{16922039} a^{7} + \frac{2812460}{16922039} a^{6} - \frac{5290571}{16922039} a^{5} - \frac{5585109}{16922039} a^{4} + \frac{497099}{16922039} a^{3} - \frac{6500202}{16922039} a^{2} - \frac{5714853}{16922039} a + \frac{3020116}{16922039}$

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:  \( 6059891910.6 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ R $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
31Data not computed