↑

Properties

Label 16.0.69052763263...1909.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 61^{9}$
Root discriminant $20.07$
Ramified primes $3, 61$
Class number $1$
Class group Trivial
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![541, -1518, 2806, -4036, 5158, -5584, 5367, -4469, 3306, -2157, 1214, -580, 242, -86, 23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 23*x^14 - 86*x^13 + 242*x^12 - 580*x^11 + 1214*x^10 - 2157*x^9 + 3306*x^8 - 4469*x^7 + 5367*x^6 - 5584*x^5 + 5158*x^4 - 4036*x^3 + 2806*x^2 - 1518*x + 541)
 
gp: K = bnfinit(x^16 - 5*x^15 + 23*x^14 - 86*x^13 + 242*x^12 - 580*x^11 + 1214*x^10 - 2157*x^9 + 3306*x^8 - 4469*x^7 + 5367*x^6 - 5584*x^5 + 5158*x^4 - 4036*x^3 + 2806*x^2 - 1518*x + 541, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 23 x^{14} - 86 x^{13} + 242 x^{12} - 580 x^{11} + 1214 x^{10} - 2157 x^{9} + 3306 x^{8} - 4469 x^{7} + 5367 x^{6} - 5584 x^{5} + 5158 x^{4} - 4036 x^{3} + 2806 x^{2} - 1518 x + 541 \)

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:  \(690527632635763191909=3^{10}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.07$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11854864872211966513741} a^{15} - \frac{86543991219693092599}{1693552124601709501963} a^{14} - \frac{662821098335002837130}{11854864872211966513741} a^{13} + \frac{2165844361540244485136}{11854864872211966513741} a^{12} - \frac{4766516644792791025373}{11854864872211966513741} a^{11} + \frac{2935172815724300375025}{11854864872211966513741} a^{10} - \frac{3173833869164707811746}{11854864872211966513741} a^{9} - \frac{498260220522035910685}{11854864872211966513741} a^{8} + \frac{94426851128646973255}{382414995877805371411} a^{7} - \frac{4742186808966383242199}{11854864872211966513741} a^{6} + \frac{5818455987099103949520}{11854864872211966513741} a^{5} + \frac{4617893581774444476739}{11854864872211966513741} a^{4} - \frac{356822767182758668397}{1693552124601709501963} a^{3} + \frac{67169937738375173418}{11854864872211966513741} a^{2} - \frac{161877398855140333532}{1693552124601709501963} a + \frac{1257430400169154863317}{11854864872211966513741}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{634897293422}{137618210046601} a^{15} - \frac{2250956409501}{137618210046601} a^{14} + \frac{10324812822366}{137618210046601} a^{13} - \frac{35261624886962}{137618210046601} a^{12} + \frac{82538074053436}{137618210046601} a^{11} - \frac{177348854883403}{137618210046601} a^{10} + \frac{328123744078536}{137618210046601} a^{9} - \frac{473916322872260}{137618210046601} a^{8} + \frac{594407288967813}{137618210046601} a^{7} - \frac{671213091605878}{137618210046601} a^{6} + \frac{668054322987216}{137618210046601} a^{5} - \frac{520812330621509}{137618210046601} a^{4} + \frac{462892941086276}{137618210046601} a^{3} - \frac{249277775900956}{137618210046601} a^{2} + \frac{205644341617454}{137618210046601} a + \frac{11754687542333}{137618210046601} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23581.2211321 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1, 8.0.18385461.1

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 $16$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ $16$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.8.6.2$x^{8} + 183 x^{4} + 14884$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$