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Properties

Label 16.4.27860145181...2944.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{44}\cdot 3^{8}\cdot 17^{6}$
Root discriminant $33.71$
Ramified primes $2, 3, 17$
Class number $2$
Class group $[2]$
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 0, 3468, 0, 2142, 0, -3128, 0, -2225, 0, -144, 0, 106, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 20*x^14 + 106*x^12 - 144*x^10 - 2225*x^8 - 3128*x^6 + 2142*x^4 + 3468*x^2 + 289)
 
gp: K = bnfinit(x^16 + 20*x^14 + 106*x^12 - 144*x^10 - 2225*x^8 - 3128*x^6 + 2142*x^4 + 3468*x^2 + 289, 1)
 

Normalized defining polynomial

\( x^{16} + 20 x^{14} + 106 x^{12} - 144 x^{10} - 2225 x^{8} - 3128 x^{6} + 2142 x^{4} + 3468 x^{2} + 289 \)

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:  \(2786014518176517344722944=2^{44}\cdot 3^{8}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.71$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} + \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} + \frac{3}{8} a$, $\frac{1}{8} a^{10} + \frac{3}{8} a^{6} - \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{11} + \frac{3}{8} a^{7} - \frac{1}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{136} a^{12} + \frac{3}{136} a^{10} + \frac{1}{34} a^{8} + \frac{43}{136} a^{6} - \frac{33}{68} a^{4} - \frac{3}{8} a^{2} + \frac{1}{8}$, $\frac{1}{136} a^{13} + \frac{3}{136} a^{11} + \frac{1}{34} a^{9} + \frac{43}{136} a^{7} - \frac{33}{68} a^{5} - \frac{3}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{1073303024} a^{14} - \frac{1}{272} a^{13} - \frac{81701}{31567736} a^{12} - \frac{3}{272} a^{11} + \frac{19630337}{1073303024} a^{10} - \frac{1}{68} a^{9} - \frac{35909103}{1073303024} a^{8} + \frac{93}{272} a^{7} + \frac{469179031}{1073303024} a^{6} + \frac{33}{136} a^{5} - \frac{370973819}{1073303024} a^{4} + \frac{3}{16} a^{3} + \frac{1710479}{7891934} a^{2} + \frac{7}{16} a - \frac{17668743}{63135472}$, $\frac{1}{1073303024} a^{15} + \frac{1168133}{1073303024} a^{13} - \frac{1}{272} a^{12} + \frac{15734119}{536651512} a^{11} - \frac{3}{272} a^{10} - \frac{20125235}{1073303024} a^{9} - \frac{1}{68} a^{8} + \frac{25551025}{268325756} a^{7} + \frac{93}{272} a^{6} + \frac{441895383}{1073303024} a^{5} + \frac{33}{136} a^{4} + \frac{1845931}{63135472} a^{3} + \frac{3}{16} a^{2} + \frac{1115310}{3945967} a + \frac{7}{16}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 442656.182557 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$