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Properties

Label 16.0.45798768824...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 17^{15}$
Root discriminant $95.24$
Ramified primes $2, 5, 17$
Class number $51472$ (GRH)
Class group $[2, 2, 12868]$ (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![10625, 0, 690625, 0, 1632000, 0, 1213375, 0, 329375, 0, 42075, 0, 2720, 0, 85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 85*x^14 + 2720*x^12 + 42075*x^10 + 329375*x^8 + 1213375*x^6 + 1632000*x^4 + 690625*x^2 + 10625)
 
gp: K = bnfinit(x^16 + 85*x^14 + 2720*x^12 + 42075*x^10 + 329375*x^8 + 1213375*x^6 + 1632000*x^4 + 690625*x^2 + 10625, 1)
 

Normalized defining polynomial

\( x^{16} + 85 x^{14} + 2720 x^{12} + 42075 x^{10} + 329375 x^{8} + 1213375 x^{6} + 1632000 x^{4} + 690625 x^{2} + 10625 \)

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:  \(45798768824157052688000000000000=2^{16}\cdot 5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.24$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(9,·)$, $\chi_{340}(267,·)$, $\chi_{340}(207,·)$, $\chi_{340}(81,·)$, $\chi_{340}(21,·)$, $\chi_{340}(23,·)$, $\chi_{340}(143,·)$, $\chi_{340}(229,·)$, $\chi_{340}(163,·)$, $\chi_{340}(101,·)$, $\chi_{340}(167,·)$, $\chi_{340}(107,·)$, $\chi_{340}(49,·)$, $\chi_{340}(283,·)$, $\chi_{340}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{325} a^{10} + \frac{1}{325} a^{8} - \frac{1}{13} a^{6} - \frac{1}{13} a^{4} - \frac{1}{13} a^{2} - \frac{1}{13}$, $\frac{1}{325} a^{11} + \frac{1}{325} a^{9} - \frac{1}{13} a^{7} - \frac{1}{13} a^{5} - \frac{1}{13} a^{3} - \frac{1}{13} a$, $\frac{1}{1625} a^{12} - \frac{5}{13}$, $\frac{1}{1625} a^{13} - \frac{5}{13} a$, $\frac{1}{208563977375} a^{14} - \frac{10523442}{208563977375} a^{12} - \frac{56118554}{41712795475} a^{10} + \frac{750469969}{41712795475} a^{8} + \frac{618541641}{8342559095} a^{6} + \frac{104744877}{8342559095} a^{4} - \frac{29949173}{1668511819} a^{2} + \frac{342631140}{1668511819}$, $\frac{1}{208563977375} a^{15} - \frac{10523442}{208563977375} a^{13} - \frac{56118554}{41712795475} a^{11} + \frac{750469969}{41712795475} a^{9} + \frac{618541641}{8342559095} a^{7} + \frac{104744877}{8342559095} a^{5} - \frac{29949173}{1668511819} a^{3} + \frac{342631140}{1668511819} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12868}$, which has order $51472$ (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:  \( 81485.0410293661 \) (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, 8.8.256461670625.1

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 $16$ R $16$ $16$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
5Data not computed
17Data not computed