↑

Properties

Label 16.0.83562110309...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{12}\cdot 11^{8}\cdot 29^{6}$
Root discriminant $131.86$
Ramified primes $2, 5, 11, 29$
Class number $128$ (GRH)
Class group $[2, 4, 16]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![336400, 0, 2018400, 0, 3490440, 0, -773720, 0, 69261, 0, 3692, 0, 119, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 + 119*x^12 + 3692*x^10 + 69261*x^8 - 773720*x^6 + 3490440*x^4 + 2018400*x^2 + 336400)
 
gp: K = bnfinit(x^16 - 32*x^14 + 119*x^12 + 3692*x^10 + 69261*x^8 - 773720*x^6 + 3490440*x^4 + 2018400*x^2 + 336400, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} + 119 x^{12} + 3692 x^{10} + 69261 x^{8} - 773720 x^{6} + 3490440 x^{4} + 2018400 x^{2} + 336400 \)

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:  \(8356211030901640462336000000000000=2^{28}\cdot 5^{12}\cdot 11^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.86$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{12} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{5}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{5}{12} a^{7} - \frac{1}{6} a^{5} - \frac{1}{4} a^{3} + \frac{1}{6} a$, $\frac{1}{6960} a^{12} - \frac{2}{435} a^{10} + \frac{1859}{6960} a^{8} - \frac{79}{580} a^{6} + \frac{2561}{6960} a^{4} + \frac{1}{6} a^{2} + \frac{1}{12}$, $\frac{1}{13920} a^{13} - \frac{1}{435} a^{11} + \frac{1859}{13920} a^{9} - \frac{79}{1160} a^{7} - \frac{4399}{13920} a^{5} - \frac{5}{12} a^{3} - \frac{11}{24} a$, $\frac{1}{841426692716599527360} a^{14} + \frac{8138495554359571}{140237782119433254560} a^{12} + \frac{33342770726101785523}{841426692716599527360} a^{10} - \frac{1}{2} a^{9} - \frac{104283564122424064543}{420713346358299763680} a^{8} + \frac{210576452311236999257}{841426692716599527360} a^{6} - \frac{1}{2} a^{5} - \frac{33814198418442194377}{140237782119433254560} a^{4} - \frac{102761668342821331}{1450735677097585392} a^{2} - \frac{1}{2} a - \frac{313863644567622047}{725367838548792696}$, $\frac{1}{8414266927165995273600} a^{15} + \frac{145310126421210829}{4207133463582997636800} a^{13} - \frac{1}{13920} a^{12} - \frac{44513377278135297181}{8414266927165995273600} a^{11} + \frac{1}{435} a^{10} - \frac{684408683887757302193}{1402377821194332545600} a^{9} - \frac{1859}{13920} a^{8} - \frac{509472022088197883639}{8414266927165995273600} a^{7} + \frac{79}{1160} a^{6} - \frac{37542614966108875609}{280475564238866509120} a^{5} + \frac{4399}{13920} a^{4} + \frac{2182147839451481683}{4835785590325284640} a^{3} + \frac{5}{12} a^{2} - \frac{497993432042800027}{1450735677097585392} a + \frac{11}{24}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{16}$, which has order $128$ (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:  \( 555036871.779 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), 4.0.7018000.3, 4.0.3625.1, \(\Q(\sqrt{5}, \sqrt{11})\), 8.0.49252324000000.63

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.30$x^{8} + 8 x^{7} + 20$$4$$2$$16$$C_2^3: C_4$$[2, 2, 3]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$