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Properties

Label 16.10.2490368000...0000.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,2^{32}\cdot 5^{15}\cdot 19$
Root discriminant $21.74$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1604

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 16, 60, -40, -350, -112, 628, 340, -465, -200, 108, -12, 30, 0, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 30*x^12 - 12*x^11 + 108*x^10 - 200*x^9 - 465*x^8 + 340*x^7 + 628*x^6 - 112*x^5 - 350*x^4 - 40*x^3 + 60*x^2 + 16*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 30*x^12 - 12*x^11 + 108*x^10 - 200*x^9 - 465*x^8 + 340*x^7 + 628*x^6 - 112*x^5 - 350*x^4 - 40*x^3 + 60*x^2 + 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 30 x^{12} - 12 x^{11} + 108 x^{10} - 200 x^{9} - 465 x^{8} + 340 x^{7} + 628 x^{6} - 112 x^{5} - 350 x^{4} - 40 x^{3} + 60 x^{2} + 16 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2490368000000000000000=-\,2^{32}\cdot 5^{15}\cdot 19\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.74$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{20} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{4} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a + \frac{1}{20}$, $\frac{1}{20} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{3}{20} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{3}{20} a - \frac{2}{5}$, $\frac{1}{20} a^{10} + \frac{1}{4} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{11} + \frac{1}{4} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{5} a$, $\frac{1}{20} a^{12} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{2} - \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{3}{10} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2} - \frac{1}{20} a + \frac{2}{5}$, $\frac{1}{40} a^{14} - \frac{1}{40} a^{13} - \frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{1}{40} a^{8} + \frac{1}{8} a^{7} - \frac{13}{40} a^{6} + \frac{1}{4} a^{5} + \frac{9}{40} a^{4} + \frac{1}{8} a^{3} + \frac{13}{40} a + \frac{1}{40}$, $\frac{1}{2940200} a^{15} + \frac{73}{9640} a^{14} + \frac{23}{2410} a^{13} + \frac{7}{9640} a^{12} + \frac{12023}{588040} a^{11} - \frac{22821}{1470100} a^{10} - \frac{9591}{588040} a^{9} - \frac{5891}{588040} a^{8} + \frac{225467}{588040} a^{7} + \frac{17033}{73505} a^{6} + \frac{1104223}{2940200} a^{5} + \frac{4041}{588040} a^{4} + \frac{26244}{73505} a^{3} + \frac{56513}{117608} a^{2} - \frac{264249}{588040} a - \frac{18748}{367525}$

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:  $12$
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:  \( 124863.087177 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1604:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1604 are not computed
Character table for t16n1604 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$