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Properties

Label 16.0.71220393711...8288.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{10}\cdot 193^{3}$
Root discriminant $15.08$
Ramified primes $2, 3, 193$
Class number $2$
Class group $[2]$
Galois group 16T1472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 38, -232, 598, -768, 392, 242, -518, 298, 14, -96, 31, 4, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 4*x^13 + 31*x^12 - 96*x^11 + 14*x^10 + 298*x^9 - 518*x^8 + 242*x^7 + 392*x^6 - 768*x^5 + 598*x^4 - 232*x^3 + 38*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 4*x^13 + 31*x^12 - 96*x^11 + 14*x^10 + 298*x^9 - 518*x^8 + 242*x^7 + 392*x^6 - 768*x^5 + 598*x^4 - 232*x^3 + 38*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 4 x^{13} + 31 x^{12} - 96 x^{11} + 14 x^{10} + 298 x^{9} - 518 x^{8} + 242 x^{7} + 392 x^{6} - 768 x^{5} + 598 x^{4} - 232 x^{3} + 38 x^{2} - 2 x + 1 \)

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:  \(7122039371137548288=2^{24}\cdot 3^{10}\cdot 193^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.08$
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:  $2, 3, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{35766799197} a^{15} + \frac{827105618}{35766799197} a^{14} - \frac{1610625887}{11922266399} a^{13} - \frac{1823805666}{11922266399} a^{12} + \frac{5446369086}{11922266399} a^{11} - \frac{7779042659}{35766799197} a^{10} - \frac{16053607717}{35766799197} a^{9} - \frac{14679632743}{35766799197} a^{8} - \frac{3528863201}{11922266399} a^{7} + \frac{5621638450}{11922266399} a^{6} + \frac{2977586439}{11922266399} a^{5} - \frac{2970852422}{35766799197} a^{4} + \frac{15141918427}{35766799197} a^{3} - \frac{755785297}{11922266399} a^{2} - \frac{3415361824}{11922266399} a - \frac{1840829994}{11922266399}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{20524857644}{35766799197} a^{15} + \frac{18121084394}{11922266399} a^{14} + \frac{13362825220}{11922266399} a^{13} - \frac{14563505024}{11922266399} a^{12} - \frac{721831366126}{35766799197} a^{11} + \frac{1000889086351}{35766799197} a^{10} + \frac{448265155807}{11922266399} a^{9} - \frac{4513556998771}{35766799197} a^{8} + \frac{3823484350913}{35766799197} a^{7} + \frac{1558795066789}{35766799197} a^{6} - \frac{6364667027500}{35766799197} a^{5} + \frac{5832042814582}{35766799197} a^{4} - \frac{2536950267352}{35766799197} a^{3} + \frac{645123610484}{35766799197} a^{2} - \frac{127597130089}{35766799197} a - \frac{19969890740}{35766799197} \) (order $12$)
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:  \( 1827.33799728 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1472 are not computed
Character table for t16n1472 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.4002048.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$193$$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.4.3.2$x^{4} - 4825$$4$$1$$3$$C_4$$[\ ]_{4}$