↑

Properties

Label 16.4.13630914358...0000.4
Degree $16$
Signature $[4, 6]$
Discriminant $2^{12}\cdot 3^{4}\cdot 5^{10}\cdot 29^{10}$
Root discriminant $49.65$
Ramified primes $2, 3, 5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1435

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![841, 0, 1682, 0, 348, 0, -1276, 0, -1675, 0, -916, 0, -132, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 132*x^12 - 916*x^10 - 1675*x^8 - 1276*x^6 + 348*x^4 + 1682*x^2 + 841)
 
gp: K = bnfinit(x^16 + 2*x^14 - 132*x^12 - 916*x^10 - 1675*x^8 - 1276*x^6 + 348*x^4 + 1682*x^2 + 841, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 132 x^{12} - 916 x^{10} - 1675 x^{8} - 1276 x^{6} + 348 x^{4} + 1682 x^{2} + 841 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1363091435892651240000000000=2^{12}\cdot 3^{4}\cdot 5^{10}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.65$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{2} + \frac{1}{20}$, $\frac{1}{20} a^{9} - \frac{1}{5} a^{7} + \frac{1}{20} a^{5} - \frac{1}{5} a^{3} + \frac{1}{20} a$, $\frac{1}{20} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{3}{10}$, $\frac{1}{20} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{3}{10} a$, $\frac{1}{17400} a^{12} + \frac{379}{17400} a^{10} + \frac{1}{174} a^{8} - \frac{63}{1160} a^{6} + \frac{83}{870} a^{4} + \frac{191}{600} a^{2} + \frac{179}{600}$, $\frac{1}{17400} a^{13} + \frac{379}{17400} a^{11} + \frac{1}{174} a^{9} - \frac{63}{1160} a^{7} + \frac{83}{870} a^{5} + \frac{191}{600} a^{3} + \frac{179}{600} a$, $\frac{1}{1009200} a^{14} - \frac{1}{34800} a^{13} + \frac{1}{504600} a^{12} + \frac{491}{34800} a^{11} + \frac{19907}{1009200} a^{10} + \frac{77}{3480} a^{9} - \frac{73}{201840} a^{8} - \frac{459}{2320} a^{7} - \frac{48649}{201840} a^{6} - \frac{79}{3480} a^{5} - \frac{8369}{34800} a^{4} + \frac{439}{1200} a^{3} - \frac{4579}{17400} a^{2} - \frac{329}{1200} a - \frac{287}{1200}$, $\frac{1}{1009200} a^{15} - \frac{9}{336400} a^{13} - \frac{1}{34800} a^{12} - \frac{2719}{168200} a^{11} + \frac{491}{34800} a^{10} + \frac{4393}{201840} a^{9} + \frac{77}{3480} a^{8} - \frac{19061}{100920} a^{7} - \frac{459}{2320} a^{6} + \frac{2747}{11600} a^{5} - \frac{79}{3480} a^{4} + \frac{4091}{11600} a^{3} + \frac{439}{1200} a^{2} + \frac{43}{150} a - \frac{329}{1200}$

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:  $9$
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:  \( 31847167.3683 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1435:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 56 conjugacy class representatives for t16n1435 are not computed
Character table for t16n1435 is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.19892278125.2

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$