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Properties

Label 16.4.33319431483...6064.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{60}\cdot 17^{2}$
Root discriminant $19.17$
Ramified primes $2, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1720

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -16, 48, -48, -76, 272, -280, -8, 285, -240, 4, 128, -82, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 82*x^12 + 128*x^11 + 4*x^10 - 240*x^9 + 285*x^8 - 8*x^7 - 280*x^6 + 272*x^5 - 76*x^4 - 48*x^3 + 48*x^2 - 16*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 82*x^12 + 128*x^11 + 4*x^10 - 240*x^9 + 285*x^8 - 8*x^7 - 280*x^6 + 272*x^5 - 76*x^4 - 48*x^3 + 48*x^2 - 16*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 82 x^{12} + 128 x^{11} + 4 x^{10} - 240 x^{9} + 285 x^{8} - 8 x^{7} - 280 x^{6} + 272 x^{5} - 76 x^{4} - 48 x^{3} + 48 x^{2} - 16 x + 2 \)

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:  \(333194314831378776064=2^{60}\cdot 17^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.17$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{391} a^{14} - \frac{7}{391} a^{13} + \frac{2}{23} a^{12} - \frac{113}{391} a^{11} + \frac{128}{391} a^{10} - \frac{162}{391} a^{9} + \frac{8}{17} a^{8} + \frac{61}{391} a^{7} - \frac{91}{391} a^{6} + \frac{9}{391} a^{5} + \frac{164}{391} a^{4} - \frac{157}{391} a^{3} + \frac{83}{391} a^{2} - \frac{134}{391} a + \frac{93}{391}$, $\frac{1}{391} a^{15} - \frac{15}{391} a^{13} + \frac{125}{391} a^{12} + \frac{7}{23} a^{11} - \frac{48}{391} a^{10} - \frac{168}{391} a^{9} + \frac{176}{391} a^{8} - \frac{55}{391} a^{7} + \frac{154}{391} a^{6} - \frac{164}{391} a^{5} - \frac{182}{391} a^{4} + \frac{157}{391} a^{3} + \frac{56}{391} a^{2} - \frac{63}{391} a - \frac{131}{391}$

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:  $9$
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:  \( 28987.6219824 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1720:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1720 are not computed
Character table for t16n1720 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.6.1073741824.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.30.46$x^{8} + 8 x^{7} + 20 x^{4} + 2$$8$$1$$30$$((C_8 : C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
2.8.30.46$x^{8} + 8 x^{7} + 20 x^{4} + 2$$8$$1$$30$$((C_8 : C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$