↑

Properties

Label 16.0.13730680642...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{4}\cdot 89^{6}$
Root discriminant $27.93$
Ramified primes $5, 29, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T211)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1871, -2516, 1061, 424, -1229, 998, -270, -88, 669, -518, 284, -44, -42, 14, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 10*x^14 + 14*x^13 - 42*x^12 - 44*x^11 + 284*x^10 - 518*x^9 + 669*x^8 - 88*x^7 - 270*x^6 + 998*x^5 - 1229*x^4 + 424*x^3 + 1061*x^2 - 2516*x + 1871)
 
gp: K = bnfinit(x^16 - 6*x^15 + 10*x^14 + 14*x^13 - 42*x^12 - 44*x^11 + 284*x^10 - 518*x^9 + 669*x^8 - 88*x^7 - 270*x^6 + 998*x^5 - 1229*x^4 + 424*x^3 + 1061*x^2 - 2516*x + 1871, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 10 x^{14} + 14 x^{13} - 42 x^{12} - 44 x^{11} + 284 x^{10} - 518 x^{9} + 669 x^{8} - 88 x^{7} - 270 x^{6} + 998 x^{5} - 1229 x^{4} + 424 x^{3} + 1061 x^{2} - 2516 x + 1871 \)

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:  \(137306806426635562890625=5^{8}\cdot 29^{4}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.93$
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:  $5, 29, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{818} a^{14} + \frac{45}{818} a^{13} - \frac{76}{409} a^{12} - \frac{51}{409} a^{11} + \frac{60}{409} a^{10} - \frac{81}{409} a^{9} - \frac{79}{409} a^{8} + \frac{45}{409} a^{7} + \frac{7}{818} a^{6} + \frac{204}{409} a^{5} - \frac{171}{409} a^{4} + \frac{164}{409} a^{3} + \frac{165}{818} a^{2} + \frac{42}{409} a + \frac{351}{818}$, $\frac{1}{85940209236435422397058} a^{15} - \frac{784712450793886371}{85940209236435422397058} a^{14} - \frac{4827746520334317458915}{42970104618217711198529} a^{13} - \frac{18174414039289882613117}{85940209236435422397058} a^{12} + \frac{8063879958827022329247}{85940209236435422397058} a^{11} - \frac{9907331022601258728757}{85940209236435422397058} a^{10} - \frac{4827534662498209148213}{85940209236435422397058} a^{9} + \frac{3551498962196303810588}{42970104618217711198529} a^{8} - \frac{9664524717739611071737}{42970104618217711198529} a^{7} + \frac{11273428493883880236576}{42970104618217711198529} a^{6} - \frac{32553381596048571876485}{85940209236435422397058} a^{5} - \frac{1807754175703600525594}{42970104618217711198529} a^{4} + \frac{9603593531143862839616}{42970104618217711198529} a^{3} + \frac{24305727851059785105475}{85940209236435422397058} a^{2} + \frac{1709314988706921148827}{85940209236435422397058} a + \frac{16789960023150053885707}{85940209236435422397058}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 13998.0176198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T211):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 4.4.64525.1, 4.4.725.1, 8.0.370549330625.2, 8.0.440605625.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
$89$89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$