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Properties

Label 16.8.29561606196...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{8}\cdot 29^{6}\cdot 89^{6}$
Root discriminant $60.18$
Ramified primes $2, 5, 29, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-225739, 419447, -218410, 26287, 43879, -38647, 10632, -5579, 1468, 752, -575, 364, -22, -9, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 - 9*x^13 - 22*x^12 + 364*x^11 - 575*x^10 + 752*x^9 + 1468*x^8 - 5579*x^7 + 10632*x^6 - 38647*x^5 + 43879*x^4 + 26287*x^3 - 218410*x^2 + 419447*x - 225739)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 - 9*x^13 - 22*x^12 + 364*x^11 - 575*x^10 + 752*x^9 + 1468*x^8 - 5579*x^7 + 10632*x^6 - 38647*x^5 + 43879*x^4 + 26287*x^3 - 218410*x^2 + 419447*x - 225739, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} - 9 x^{13} - 22 x^{12} + 364 x^{11} - 575 x^{10} + 752 x^{9} + 1468 x^{8} - 5579 x^{7} + 10632 x^{6} - 38647 x^{5} + 43879 x^{4} + 26287 x^{3} - 218410 x^{2} + 419447 x - 225739 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29561606196428930148100000000=2^{8}\cdot 5^{8}\cdot 29^{6}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.18$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} - \frac{6}{29} a^{13} + \frac{7}{29} a^{12} + \frac{12}{29} a^{11} + \frac{5}{29} a^{10} - \frac{6}{29} a^{9} + \frac{12}{29} a^{8} + \frac{9}{29} a^{7} - \frac{12}{29} a^{6} + \frac{4}{29} a^{5} - \frac{7}{29} a^{4} - \frac{9}{29} a^{3} - \frac{2}{29} a^{2} - \frac{3}{29} a - \frac{3}{29}$, $\frac{1}{134523079036742447903843484753508837} a^{15} + \frac{228407969518572045866916878421428}{134523079036742447903843484753508837} a^{14} + \frac{25761709757367085790343303934611121}{134523079036742447903843484753508837} a^{13} + \frac{3847358318403850257224330022138399}{134523079036742447903843484753508837} a^{12} + \frac{29013073966214460074066701933655575}{134523079036742447903843484753508837} a^{11} + \frac{59433640411392497802343272708891232}{134523079036742447903843484753508837} a^{10} + \frac{54722447829750692191173439132216959}{134523079036742447903843484753508837} a^{9} + \frac{32647028765772936477424980490539928}{134523079036742447903843484753508837} a^{8} - \frac{36131006772388015776757313223651157}{134523079036742447903843484753508837} a^{7} - \frac{1741626919487360078504798088984096}{4638726863335946479442878784603753} a^{6} - \frac{46197615301909526030017427643988885}{134523079036742447903843484753508837} a^{5} - \frac{14002564452600674140538379358898596}{134523079036742447903843484753508837} a^{4} + \frac{29829017112806577832176565116969744}{134523079036742447903843484753508837} a^{3} - \frac{60723127688968195725749451719986268}{134523079036742447903843484753508837} a^{2} + \frac{38493424184890850862483752153243602}{134523079036742447903843484753508837} a - \frac{35407040200166279232449144161644096}{134523079036742447903843484753508837}$

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:  $11$
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:  \( 81517515.756 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.4.725.1, 4.4.2225.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$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$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$