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Properties

Label 16.0.11181340044...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 17^{15}$
Root discriminant $31.84$
Ramified primes $5, 17$
Class number $34$ (GRH)
Class group $[34]$ (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![3571, -3554, 3554, -3350, 3350, -2636, 2636, -1514, 1514, -579, 579, -137, 137, -18, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 18*x^14 - 18*x^13 + 137*x^12 - 137*x^11 + 579*x^10 - 579*x^9 + 1514*x^8 - 1514*x^7 + 2636*x^6 - 2636*x^5 + 3350*x^4 - 3350*x^3 + 3554*x^2 - 3554*x + 3571)
 
gp: K = bnfinit(x^16 - x^15 + 18*x^14 - 18*x^13 + 137*x^12 - 137*x^11 + 579*x^10 - 579*x^9 + 1514*x^8 - 1514*x^7 + 2636*x^6 - 2636*x^5 + 3350*x^4 - 3350*x^3 + 3554*x^2 - 3554*x + 3571, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 18 x^{14} - 18 x^{13} + 137 x^{12} - 137 x^{11} + 579 x^{10} - 579 x^{9} + 1514 x^{8} - 1514 x^{7} + 2636 x^{6} - 2636 x^{5} + 3350 x^{4} - 3350 x^{3} + 3554 x^{2} - 3554 x + 3571 \)

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:  \(1118134004496021794140625=5^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.84$
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$$\chi_{85}(1,·)$, $\chi_{85}(66,·)$, $\chi_{85}(74,·)$, $\chi_{85}(76,·)$, $\chi_{85}(14,·)$, $\chi_{85}(79,·)$, $\chi_{85}(16,·)$, $\chi_{85}(81,·)$, $\chi_{85}(21,·)$, $\chi_{85}(24,·)$, $\chi_{85}(26,·)$, $\chi_{85}(29,·)$, $\chi_{85}(36,·)$, $\chi_{85}(39,·)$, $\chi_{85}(44,·)$, $\chi_{85}(54,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1597} a^{9} - \frac{610}{1597} a^{8} + \frac{9}{1597} a^{7} - \frac{89}{1597} a^{6} + \frac{27}{1597} a^{5} + \frac{576}{1597} a^{4} + \frac{30}{1597} a^{3} - \frac{178}{1597} a^{2} + \frac{9}{1597} a + \frac{377}{1597}$, $\frac{1}{1597} a^{10} + \frac{10}{1597} a^{8} + \frac{610}{1597} a^{7} + \frac{35}{1597} a^{6} - \frac{521}{1597} a^{5} + \frac{50}{1597} a^{4} + \frac{555}{1597} a^{3} + \frac{25}{1597} a^{2} - \frac{521}{1597} a + \frac{2}{1597}$, $\frac{1}{1597} a^{11} + \frac{322}{1597} a^{8} - \frac{55}{1597} a^{7} + \frac{369}{1597} a^{6} - \frac{220}{1597} a^{5} - \frac{414}{1597} a^{4} - \frac{275}{1597} a^{3} - \frac{338}{1597} a^{2} - \frac{88}{1597} a - \frac{576}{1597}$, $\frac{1}{1597} a^{12} - \frac{66}{1597} a^{8} + \frac{665}{1597} a^{7} - \frac{308}{1597} a^{6} + \frac{474}{1597} a^{5} - \frac{495}{1597} a^{4} - \frac{416}{1597} a^{3} - \frac{264}{1597} a^{2} - \frac{280}{1597} a - \frac{22}{1597}$, $\frac{1}{1597} a^{13} + \frac{330}{1597} a^{8} + \frac{286}{1597} a^{7} - \frac{609}{1597} a^{6} - \frac{310}{1597} a^{5} - \frac{728}{1597} a^{4} + \frac{119}{1597} a^{3} + \frac{748}{1597} a^{2} + \frac{572}{1597} a - \frac{670}{1597}$, $\frac{1}{1597} a^{14} + \frac{364}{1597} a^{8} - \frac{385}{1597} a^{7} + \frac{314}{1597} a^{6} - \frac{56}{1597} a^{5} + \frac{82}{1597} a^{4} + \frac{430}{1597} a^{3} + \frac{223}{1597} a^{2} - \frac{446}{1597} a + \frac{156}{1597}$, $\frac{1}{1597} a^{15} - \frac{328}{1597} a^{8} + \frac{232}{1597} a^{7} + \frac{400}{1597} a^{6} - \frac{164}{1597} a^{5} - \frac{27}{1597} a^{4} + \frac{482}{1597} a^{3} + \frac{466}{1597} a^{2} + \frac{74}{1597} a + \frac{114}{1597}$

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

Class group and class number

$C_{34}$, which has order $34$ (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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.

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