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Properties

Label 16.0.74163788185...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{8}\cdot 29^{4}$
Root discriminant $17.45$
Ramified primes $2, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![116, 424, 224, -456, 132, 568, -432, 132, 258, -388, 380, -244, 128, -48, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 18*x^14 - 48*x^13 + 128*x^12 - 244*x^11 + 380*x^10 - 388*x^9 + 258*x^8 + 132*x^7 - 432*x^6 + 568*x^5 + 132*x^4 - 456*x^3 + 224*x^2 + 424*x + 116)
 
gp: K = bnfinit(x^16 - 4*x^15 + 18*x^14 - 48*x^13 + 128*x^12 - 244*x^11 + 380*x^10 - 388*x^9 + 258*x^8 + 132*x^7 - 432*x^6 + 568*x^5 + 132*x^4 - 456*x^3 + 224*x^2 + 424*x + 116, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 18 x^{14} - 48 x^{13} + 128 x^{12} - 244 x^{11} + 380 x^{10} - 388 x^{9} + 258 x^{8} + 132 x^{7} - 432 x^{6} + 568 x^{5} + 132 x^{4} - 456 x^{3} + 224 x^{2} + 424 x + 116 \)

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:  \(74163788185600000000=2^{28}\cdot 5^{8}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.45$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{58} a^{14} - \frac{3}{29} a^{13} - \frac{13}{58} a^{11} + \frac{9}{58} a^{10} + \frac{6}{29} a^{9} - \frac{1}{58} a^{8} + \frac{4}{29} a^{7} - \frac{9}{29} a^{6} - \frac{7}{29} a^{5} + \frac{10}{29} a^{4} + \frac{10}{29} a^{3} + \frac{7}{29} a^{2} + \frac{9}{29} a$, $\frac{1}{63191668993639162} a^{15} - \frac{410787080634737}{63191668993639162} a^{14} - \frac{3872220168652483}{63191668993639162} a^{13} - \frac{8622877889919865}{63191668993639162} a^{12} + \frac{104004681703439}{1089511534373089} a^{11} - \frac{855908126475125}{63191668993639162} a^{10} - \frac{3099923831029876}{31595834496819581} a^{9} + \frac{5148423184874285}{31595834496819581} a^{8} + \frac{15647223239379120}{31595834496819581} a^{7} - \frac{2429618195420433}{31595834496819581} a^{6} + \frac{5756352966463618}{31595834496819581} a^{5} - \frac{5718002993126983}{31595834496819581} a^{4} - \frac{962739876852923}{31595834496819581} a^{3} - \frac{15401258990495714}{31595834496819581} a^{2} - \frac{8598312597548670}{31595834496819581} a - \frac{144951045531807}{1089511534373089}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{252896874589}{5175826766618} a^{15} + \frac{1148732357479}{5175826766618} a^{14} - \frac{2582693501169}{2587913383309} a^{13} + \frac{14919660532035}{5175826766618} a^{12} - \frac{20178072825306}{2587913383309} a^{11} + \frac{41712825439867}{2587913383309} a^{10} - \frac{70446749908765}{2587913383309} a^{9} + \frac{87101954435550}{2587913383309} a^{8} - \frac{79945485647769}{2587913383309} a^{7} + \frac{27621870151646}{2587913383309} a^{6} + \frac{38064150419388}{2587913383309} a^{5} - \frac{89709422636924}{2587913383309} a^{4} + \frac{30631903604870}{2587913383309} a^{3} + \frac{41905840921956}{2587913383309} a^{2} - \frac{48383202988664}{2587913383309} a - \frac{27490498625391}{2587913383309} \) (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:  \( 3332.31310139 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.400.1 x2, 4.0.320.1 x2, \(\Q(i, \sqrt{5})\), 8.0.2560000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$