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Properties

Label 16.4.96143571830...0304.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{44}\cdot 3^{8}\cdot 97^{6}$
Root discriminant $64.78$
Ramified primes $2, 3, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9409, 0, 75272, 0, 74302, 0, -13192, 0, -19062, 0, -3080, 0, 142, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 142*x^12 - 3080*x^10 - 19062*x^8 - 13192*x^6 + 74302*x^4 + 75272*x^2 + 9409)
 
gp: K = bnfinit(x^16 + 40*x^14 + 142*x^12 - 3080*x^10 - 19062*x^8 - 13192*x^6 + 74302*x^4 + 75272*x^2 + 9409, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} + 142 x^{12} - 3080 x^{10} - 19062 x^{8} - 13192 x^{6} + 74302 x^{4} + 75272 x^{2} + 9409 \)

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:  \(96143571830568172203282530304=2^{44}\cdot 3^{8}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.78$
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:  $2, 3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{776} a^{12} + \frac{5}{97} a^{10} + \frac{45}{776} a^{8} + \frac{3}{97} a^{6} - \frac{147}{776} a^{4} + \frac{1}{8}$, $\frac{1}{776} a^{13} + \frac{5}{97} a^{11} + \frac{45}{776} a^{9} + \frac{3}{97} a^{7} - \frac{147}{776} a^{5} + \frac{1}{8} a$, $\frac{1}{10528761771400304} a^{14} - \frac{5212619479639}{10528761771400304} a^{12} - \frac{1}{4} a^{11} + \frac{1422690635216957}{10528761771400304} a^{10} - \frac{1}{4} a^{9} + \frac{2439359570465861}{10528761771400304} a^{8} - \frac{4910691015799971}{10528761771400304} a^{6} - \frac{3024199011627195}{10528761771400304} a^{4} - \frac{1}{4} a^{3} - \frac{33580906879823}{108543935787632} a^{2} - \frac{1}{4} a + \frac{44946606594649}{108543935787632}$, $\frac{1}{10528761771400304} a^{15} - \frac{5212619479639}{10528761771400304} a^{13} + \frac{1422690635216957}{10528761771400304} a^{11} - \frac{1}{4} a^{10} + \frac{2439359570465861}{10528761771400304} a^{9} - \frac{1}{4} a^{8} - \frac{4910691015799971}{10528761771400304} a^{7} - \frac{3024199011627195}{10528761771400304} a^{5} - \frac{33580906879823}{108543935787632} a^{3} - \frac{1}{4} a^{2} + \frac{44946606594649}{108543935787632} a - \frac{1}{4}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 263522701.543 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.13968.1, 4.4.223488.2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.49946886144.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.20.58$x^{8} + 8 x^{6} + 64 x + 16$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.24.33$x^{8} + 16 x^{4} + 144$$8$$1$$24$$Q_8:C_2$$[2, 3, 4]^{2}$
$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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.4.3.3$x^{4} + 485$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.3.4$x^{4} + 12125$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$