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Properties

Label 16.0.30953458224...3216.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{26}\cdot 3^{2}\cdot 8461^{4}$
Root discriminant $33.94$
Ramified primes $2, 3, 8461$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group 16T1665

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7569, 0, -3804, 0, 2540, 0, -684, 0, 102, 0, 44, 0, -4, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 4*x^12 + 44*x^10 + 102*x^8 - 684*x^6 + 2540*x^4 - 3804*x^2 + 7569)
 
gp: K = bnfinit(x^16 - 4*x^14 - 4*x^12 + 44*x^10 + 102*x^8 - 684*x^6 + 2540*x^4 - 3804*x^2 + 7569, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 4 x^{12} + 44 x^{10} + 102 x^{8} - 684 x^{6} + 2540 x^{4} - 3804 x^{2} + 7569 \)

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:  \(3095345822428295086473216=2^{26}\cdot 3^{2}\cdot 8461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.94$
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:  $2, 3, 8461$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{7}{16} a + \frac{5}{16}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{16} a^{2} + \frac{3}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{3}{16}$, $\frac{1}{96} a^{12} + \frac{5}{96} a^{8} + \frac{1}{24} a^{6} - \frac{1}{4} a^{5} - \frac{17}{96} a^{4} - \frac{5}{24} a^{2} + \frac{1}{4} a + \frac{9}{32}$, $\frac{1}{96} a^{13} - \frac{1}{96} a^{9} - \frac{1}{16} a^{8} - \frac{1}{12} a^{7} - \frac{1}{8} a^{6} - \frac{17}{96} a^{5} - \frac{1}{4} a^{4} + \frac{1}{6} a^{3} - \frac{1}{8} a^{2} + \frac{3}{32} a - \frac{7}{16}$, $\frac{1}{35196384} a^{14} + \frac{53269}{11732128} a^{12} - \frac{459667}{35196384} a^{10} - \frac{589937}{35196384} a^{8} + \frac{3891835}{35196384} a^{6} - \frac{6058259}{35196384} a^{4} + \frac{435005}{11732128} a^{2} - \frac{4541225}{11732128}$, $\frac{1}{2041390272} a^{15} - \frac{1}{70392768} a^{14} + \frac{786527}{680463424} a^{13} - \frac{53269}{23464256} a^{12} - \frac{4859215}{2041390272} a^{11} + \frac{459667}{70392768} a^{10} - \frac{7189259}{2041390272} a^{9} - \frac{3809611}{70392768} a^{8} - \frac{224884661}{2041390272} a^{7} - \frac{3891835}{70392768} a^{6} + \frac{431696767}{2041390272} a^{5} - \frac{2740837}{70392768} a^{4} - \frac{68491247}{680463424} a^{3} + \frac{5431059}{23464256} a^{2} + \frac{79783445}{680463424} a - \frac{8657419}{23464256}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( -\frac{233}{1099887} a^{14} + \frac{11629}{17598192} a^{12} + \frac{7315}{8799096} a^{10} - \frac{40745}{5866064} a^{8} - \frac{129523}{4399548} a^{6} + \frac{1943039}{17598192} a^{4} - \frac{5830609}{8799096} a^{2} + \frac{3525757}{5866064} \) (order $4$)
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:  \( 486888.507154 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1665:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 78 conjugacy class representatives for t16n1665 are not computed
Character table for t16n1665 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.4.33844.1, 8.0.73306645504.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.12.20.68$x^{12} + 2 x^{9} + 2$$12$$1$$20$$(C_6\times C_2):C_2$$[2, 2]_{3}^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
8461Data not computed