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Properties

Label 16.8.36582036480...0000.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{10}\cdot 5^{15}\cdot 11^{2}$
Root discriminant $34.29$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1584

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-449, -1398, -1760, -220, 4200, 5826, 82, -4570, -1495, 1610, 382, -406, 10, 60, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 60*x^13 + 10*x^12 - 406*x^11 + 382*x^10 + 1610*x^9 - 1495*x^8 - 4570*x^7 + 82*x^6 + 5826*x^5 + 4200*x^4 - 220*x^3 - 1760*x^2 - 1398*x - 449)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^14 + 60*x^13 + 10*x^12 - 406*x^11 + 382*x^10 + 1610*x^9 - 1495*x^8 - 4570*x^7 + 82*x^6 + 5826*x^5 + 4200*x^4 - 220*x^3 - 1760*x^2 - 1398*x - 449, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{14} + 60 x^{13} + 10 x^{12} - 406 x^{11} + 382 x^{10} + 1610 x^{9} - 1495 x^{8} - 4570 x^{7} + 82 x^{6} + 5826 x^{5} + 4200 x^{4} - 220 x^{3} - 1760 x^{2} - 1398 x - 449 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3658203648000000000000000=2^{24}\cdot 3^{10}\cdot 5^{15}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.29$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{1009386329350206044431756} a^{15} - \frac{47039414318952582779685}{1009386329350206044431756} a^{14} - \frac{24695724422024926448519}{252346582337551511107939} a^{13} - \frac{16901781055925803768628}{252346582337551511107939} a^{12} + \frac{138337772804885208616967}{1009386329350206044431756} a^{11} + \frac{185924301475190287820497}{1009386329350206044431756} a^{10} + \frac{127753846398786915007457}{1009386329350206044431756} a^{9} + \frac{32724343346032122764749}{1009386329350206044431756} a^{8} - \frac{357663218801943915245643}{1009386329350206044431756} a^{7} + \frac{335818269338738309249975}{1009386329350206044431756} a^{6} + \frac{124045407068334981105505}{1009386329350206044431756} a^{5} - \frac{2284221270944366337027}{1009386329350206044431756} a^{4} - \frac{54091405198513549509657}{504693164675103022215878} a^{3} + \frac{237528233250421539763649}{504693164675103022215878} a^{2} - \frac{304007752359304975147477}{1009386329350206044431756} a + \frac{44707529926673760464567}{1009386329350206044431756}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 2363000.61685 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1584:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1584 are not computed
Character table for t16n1584 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.1620000000.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$