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Properties

Label 16.0.40552519924...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 359^{4}$
Root discriminant $14.55$
Ramified primes $5, 359$
Class number $2$
Class group $[2]$
Galois group 16T1031

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 17, 26, 29, 4, 33, -21, 63, -39, 43, -22, 20, -8, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 - 8*x^13 + 20*x^12 - 22*x^11 + 43*x^10 - 39*x^9 + 63*x^8 - 21*x^7 + 33*x^6 + 4*x^5 + 29*x^4 + 26*x^3 + 17*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 - 8*x^13 + 20*x^12 - 22*x^11 + 43*x^10 - 39*x^9 + 63*x^8 - 21*x^7 + 33*x^6 + 4*x^5 + 29*x^4 + 26*x^3 + 17*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} - 8 x^{13} + 20 x^{12} - 22 x^{11} + 43 x^{10} - 39 x^{9} + 63 x^{8} - 21 x^{7} + 33 x^{6} + 4 x^{5} + 29 x^{4} + 26 x^{3} + 17 x^{2} + 4 x + 1 \)

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:  \(4055251992431640625=5^{12}\cdot 359^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.55$
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:  $5, 359$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{13} - \frac{11}{25} a^{11} + \frac{1}{25} a^{10} + \frac{11}{25} a^{9} - \frac{2}{25} a^{8} - \frac{1}{25} a^{7} + \frac{12}{25} a^{6} - \frac{9}{25} a^{5} + \frac{4}{25} a^{4} - \frac{4}{25} a^{3} + \frac{7}{25} a^{2} + \frac{2}{5} a + \frac{9}{25}$, $\frac{1}{8641225} a^{15} + \frac{86374}{8641225} a^{14} + \frac{152614}{8641225} a^{13} - \frac{174706}{8641225} a^{12} + \frac{4086394}{8641225} a^{11} + \frac{3089633}{8641225} a^{10} + \frac{15507}{1728245} a^{9} + \frac{457528}{1728245} a^{8} + \frac{213164}{1728245} a^{7} - \frac{192051}{1728245} a^{6} - \frac{3816999}{8641225} a^{5} + \frac{3995169}{8641225} a^{4} + \frac{779794}{8641225} a^{3} + \frac{939834}{8641225} a^{2} + \frac{668759}{8641225} a - \frac{133492}{8641225}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{585908}{1728245} a^{15} + \frac{6977609}{8641225} a^{14} - \frac{20524927}{8641225} a^{13} + \frac{6332923}{1728245} a^{12} - \frac{71753839}{8641225} a^{11} + \frac{92947449}{8641225} a^{10} - \frac{164209421}{8641225} a^{9} + \frac{179317742}{8641225} a^{8} - \frac{256967539}{8641225} a^{7} + \frac{162714648}{8641225} a^{6} - \frac{160180041}{8641225} a^{5} + \frac{45887746}{8641225} a^{4} - \frac{94975461}{8641225} a^{3} - \frac{41147482}{8641225} a^{2} - \frac{6306964}{1728245} a + \frac{2529431}{8641225} \) (order $10$)
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:  \( 1863.17635715 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1031:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 13 conjugacy class representatives for t16n1031
Character table for t16n1031

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.4.402753125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
359Data not computed