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Properties

Label 16.16.6557827967...7536.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 13^{12}$
Root discriminant $54.77$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 0, -4688, 0, 13832, 0, -18232, 0, 11579, 0, -3680, 0, 572, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 572*x^12 - 3680*x^10 + 11579*x^8 - 18232*x^6 + 13832*x^4 - 4688*x^2 + 529)
 
gp: K = bnfinit(x^16 - 40*x^14 + 572*x^12 - 3680*x^10 + 11579*x^8 - 18232*x^6 + 13832*x^4 - 4688*x^2 + 529, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} + 572 x^{12} - 3680 x^{10} + 11579 x^{8} - 18232 x^{6} + 13832 x^{4} - 4688 x^{2} + 529 \)

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:  \(6557827967253220516257857536=2^{48}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.77$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(208=2^{4}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{208}(1,·)$, $\chi_{208}(129,·)$, $\chi_{208}(203,·)$, $\chi_{208}(77,·)$, $\chi_{208}(83,·)$, $\chi_{208}(151,·)$, $\chi_{208}(25,·)$, $\chi_{208}(157,·)$, $\chi_{208}(31,·)$, $\chi_{208}(99,·)$, $\chi_{208}(105,·)$, $\chi_{208}(135,·)$, $\chi_{208}(47,·)$, $\chi_{208}(53,·)$, $\chi_{208}(187,·)$, $\chi_{208}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{69} a^{11} + \frac{5}{69} a^{9} + \frac{3}{23} a^{7} + \frac{10}{23} a^{5} - \frac{5}{69} a^{3} - \frac{7}{69} a$, $\frac{1}{207} a^{12} + \frac{28}{207} a^{10} + \frac{32}{207} a^{8} + \frac{10}{69} a^{6} - \frac{5}{207} a^{4} + \frac{16}{207} a^{2} + \frac{1}{9}$, $\frac{1}{207} a^{13} + \frac{1}{207} a^{11} - \frac{34}{207} a^{9} - \frac{2}{69} a^{7} + \frac{13}{207} a^{5} - \frac{56}{207} a^{3} + \frac{74}{207} a$, $\frac{1}{537993} a^{14} - \frac{167}{537993} a^{12} - \frac{1652}{23391} a^{10} - \frac{386}{2599} a^{8} - \frac{2957}{537993} a^{6} + \frac{177976}{537993} a^{4} - \frac{232522}{537993} a^{2} + \frac{35}{339}$, $\frac{1}{537993} a^{15} - \frac{167}{537993} a^{13} + \frac{43}{23391} a^{11} - \frac{932}{7797} a^{9} - \frac{190085}{537993} a^{7} - \frac{266453}{537993} a^{5} + \frac{110546}{537993} a^{3} + \frac{2048}{7797} a$

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:  \( 411384624.9155599 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{13}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{13})\), 4.4.346112.1, 4.4.4499456.2, 4.4.4499456.1, 4.4.140608.1, 4.4.35152.1, 8.8.119793516544.1, 8.8.20245104295936.1, 8.8.316329754624.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
13Data not computed