↑

Properties

Label 16.8.13096085431...1024.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{62}\cdot 7^{6}\cdot 17^{6}$
Root discriminant $88.07$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14161, 0, -94248, 0, -274012, 0, -14920, 0, 45974, 0, -3096, 0, -764, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 764*x^12 - 3096*x^10 + 45974*x^8 - 14920*x^6 - 274012*x^4 - 94248*x^2 + 14161)
 
gp: K = bnfinit(x^16 + 8*x^14 - 764*x^12 - 3096*x^10 + 45974*x^8 - 14920*x^6 - 274012*x^4 - 94248*x^2 + 14161, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 764 x^{12} - 3096 x^{10} + 45974 x^{8} - 14920 x^{6} - 274012 x^{4} - 94248 x^{2} + 14161 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13096085431976788600857793921024=2^{62}\cdot 7^{6}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.07$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{3}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{5}{16} a - \frac{5}{16}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{8} - \frac{1}{24} a^{6} + \frac{1}{24} a^{4} + \frac{7}{16} a^{2} - \frac{5}{48}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} - \frac{1}{24} a^{7} + \frac{1}{24} a^{5} + \frac{7}{16} a^{3} - \frac{5}{48} a$, $\frac{1}{288} a^{12} + \frac{1}{96} a^{8} - \frac{1}{12} a^{6} - \frac{61}{288} a^{4} + \frac{17}{36} a^{2} + \frac{97}{288}$, $\frac{1}{576} a^{13} - \frac{1}{576} a^{12} - \frac{1}{96} a^{11} - \frac{1}{96} a^{10} + \frac{1}{64} a^{9} - \frac{11}{192} a^{8} + \frac{5}{48} a^{7} - \frac{1}{16} a^{6} - \frac{1}{576} a^{5} + \frac{49}{576} a^{4} + \frac{41}{288} a^{3} + \frac{121}{288} a^{2} + \frac{199}{576} a + \frac{257}{576}$, $\frac{1}{10241443345912128} a^{14} + \frac{3236412835531}{10241443345912128} a^{12} - \frac{3757550682779}{3413814448637376} a^{10} - \frac{75812418959917}{3413814448637376} a^{8} - \frac{1044572523365821}{10241443345912128} a^{6} - \frac{513848302245965}{3413814448637376} a^{4} - \frac{4213272586481515}{10241443345912128} a^{2} + \frac{21592214035469}{86062549125312}$, $\frac{1}{10241443345912128} a^{15} + \frac{3236412835531}{10241443345912128} a^{13} - \frac{3757550682779}{3413814448637376} a^{11} - \frac{75812418959917}{3413814448637376} a^{9} - \frac{1044572523365821}{10241443345912128} a^{7} + \frac{339605309913379}{3413814448637376} a^{5} - \frac{1}{4} a^{4} + \frac{907449086474549}{10241443345912128} a^{3} - \frac{1}{2} a^{2} + \frac{76576754141}{86062549125312} a + \frac{1}{4}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 3112328064.01 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.121856.1, 4.4.243712.2, 8.8.950328623104.5

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$