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Properties

Label 16.0.12882379072...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{2}\cdot 89^{4}$
Root discriminant $15.64$
Ramified primes $5, 29, 89$
Class number $1$
Class group Trivial
Galois group $C_2^2.C_2^5.C_2$ (as 16T610)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 22, -62, 137, -225, 288, -289, 195, -76, 53, -65, 27, 2, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 2*x^13 + 27*x^12 - 65*x^11 + 53*x^10 - 76*x^9 + 195*x^8 - 289*x^7 + 288*x^6 - 225*x^5 + 137*x^4 - 62*x^3 + 22*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 2*x^13 + 27*x^12 - 65*x^11 + 53*x^10 - 76*x^9 + 195*x^8 - 289*x^7 + 288*x^6 - 225*x^5 + 137*x^4 - 62*x^3 + 22*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 2 x^{13} + 27 x^{12} - 65 x^{11} + 53 x^{10} - 76 x^{9} + 195 x^{8} - 289 x^{7} + 288 x^{6} - 225 x^{5} + 137 x^{4} - 62 x^{3} + 22 x^{2} - 6 x + 1 \)

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:  \(12882379072509765625=5^{12}\cdot 29^{2}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.64$
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:  $5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1209286021} a^{15} - \frac{86305067}{1209286021} a^{14} - \frac{8560310}{29494781} a^{13} - \frac{445596403}{1209286021} a^{12} + \frac{145198900}{1209286021} a^{11} + \frac{317768759}{1209286021} a^{10} + \frac{258816251}{1209286021} a^{9} - \frac{248264413}{1209286021} a^{8} + \frac{228743208}{1209286021} a^{7} + \frac{2557912}{29494781} a^{6} + \frac{444173132}{1209286021} a^{5} - \frac{298033142}{1209286021} a^{4} + \frac{270377652}{1209286021} a^{3} - \frac{399004121}{1209286021} a^{2} - \frac{369659125}{1209286021} a + \frac{127038651}{1209286021}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{2740353664}{1209286021} a^{15} + \frac{10045211574}{1209286021} a^{14} - \frac{47647995}{29494781} a^{13} - \frac{6664473063}{1209286021} a^{12} - \frac{76246472823}{1209286021} a^{11} + \frac{152612563348}{1209286021} a^{10} - \frac{89600085781}{1209286021} a^{9} + \frac{171621708975}{1209286021} a^{8} - \frac{471404556934}{1209286021} a^{7} + \frac{15144748011}{29494781} a^{6} - \frac{557632332323}{1209286021} a^{5} + \frac{398255996828}{1209286021} a^{4} - \frac{205752197840}{1209286021} a^{3} + \frac{73722787664}{1209286021} a^{2} - \frac{21349157511}{1209286021} a + \frac{3896984120}{1209286021} \) (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:  \( 2956.99382672 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T610):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.11125.1, 4.4.2225.1, 8.0.123765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
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$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$