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Properties

Label 16.0.61870785919...4656.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{14}\cdot 7^{6}$
Root discriminant $30.69$
Ramified primes $2, 3, 7$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4243, -22004, 45744, -47512, 21824, 4692, -10832, 3040, 2784, -2348, 328, 336, -160, -4, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 4*x^13 - 160*x^12 + 336*x^11 + 328*x^10 - 2348*x^9 + 2784*x^8 + 3040*x^7 - 10832*x^6 + 4692*x^5 + 21824*x^4 - 47512*x^3 + 45744*x^2 - 22004*x + 4243)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 4*x^13 - 160*x^12 + 336*x^11 + 328*x^10 - 2348*x^9 + 2784*x^8 + 3040*x^7 - 10832*x^6 + 4692*x^5 + 21824*x^4 - 47512*x^3 + 45744*x^2 - 22004*x + 4243, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 4 x^{13} - 160 x^{12} + 336 x^{11} + 328 x^{10} - 2348 x^{9} + 2784 x^{8} + 3040 x^{7} - 10832 x^{6} + 4692 x^{5} + 21824 x^{4} - 47512 x^{3} + 45744 x^{2} - 22004 x + 4243 \)

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:  \(618707859192665295814656=2^{40}\cdot 3^{14}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.69$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{2}{9}$, $\frac{1}{126} a^{13} - \frac{1}{63} a^{12} + \frac{1}{21} a^{11} - \frac{1}{63} a^{10} + \frac{5}{63} a^{9} + \frac{1}{21} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{11}{42} a^{5} + \frac{26}{63} a^{4} - \frac{13}{63} a^{3} + \frac{10}{21} a^{2} + \frac{20}{63} a + \frac{29}{63}$, $\frac{1}{126} a^{14} + \frac{1}{63} a^{12} + \frac{5}{63} a^{11} + \frac{1}{21} a^{10} + \frac{5}{126} a^{9} + \frac{1}{14} a^{8} + \frac{5}{21} a^{7} + \frac{1}{6} a^{6} - \frac{25}{63} a^{5} - \frac{1}{21} a^{4} - \frac{17}{63} a^{3} - \frac{25}{63} a^{2} + \frac{11}{42} a + \frac{11}{126}$, $\frac{1}{304349218838049294} a^{15} + \frac{12486353642575}{14492819944669014} a^{14} + \frac{56129923707997}{14492819944669014} a^{13} - \frac{138137215282298}{21739229917003521} a^{12} - \frac{3262707802669498}{50724869806341549} a^{11} + \frac{376049377223379}{5636096645149061} a^{10} + \frac{9565349512060907}{152174609419024647} a^{9} - \frac{1746041600460025}{101449739612683098} a^{8} - \frac{25639073727561733}{101449739612683098} a^{7} - \frac{21744690323731217}{304349218838049294} a^{6} - \frac{29795730994948327}{101449739612683098} a^{5} - \frac{12931153410668033}{50724869806341549} a^{4} + \frac{30774594973189642}{152174609419024647} a^{3} - \frac{3132344103489586}{16908289935447183} a^{2} + \frac{796843373100769}{3901913062026273} a - \frac{32949965947868939}{304349218838049294}$

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

Class group and class number

$C_{4}$, 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:  \( -\frac{137304800}{13774391001} a^{15} + \frac{897453628}{13774391001} a^{14} - \frac{287946055}{1967770143} a^{13} - \frac{734747444}{4591463667} a^{12} + \frac{2627341018}{1967770143} a^{11} - \frac{2863310152}{1967770143} a^{10} - \frac{70367704159}{13774391001} a^{9} + \frac{144947586677}{9182927334} a^{8} - \frac{27840881218}{4591463667} a^{7} - \frac{499325536190}{13774391001} a^{6} + \frac{767347352329}{13774391001} a^{5} + \frac{356387398124}{13774391001} a^{4} - \frac{791291927332}{4591463667} a^{3} + \frac{3229436293378}{13774391001} a^{2} - \frac{2024906698309}{13774391001} a + \frac{999198064057}{27548782002} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 463430.206047 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

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 $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.4032.1, 4.0.12096.1, 4.0.432.1, 8.0.2341011456.4

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$