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Properties

Label 16.8.15911238919...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{8}\cdot 89^{6}$
Root discriminant $158.53$
Ramified primes $2, 5, 29, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![140059320025, 0, 82103739325, 0, -21410182205, 0, 920268455, 0, 29111476, 0, -1116297, 0, -27457, 0, -62, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 62*x^14 - 27457*x^12 - 1116297*x^10 + 29111476*x^8 + 920268455*x^6 - 21410182205*x^4 + 82103739325*x^2 + 140059320025)
 
gp: K = bnfinit(x^16 - 62*x^14 - 27457*x^12 - 1116297*x^10 + 29111476*x^8 + 920268455*x^6 - 21410182205*x^4 + 82103739325*x^2 + 140059320025, 1)
 

Normalized defining polynomial

\( x^{16} - 62 x^{14} - 27457 x^{12} - 1116297 x^{10} + 29111476 x^{8} + 920268455 x^{6} - 21410182205 x^{4} + 82103739325 x^{2} + 140059320025 \)

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:  \(159112389191659073629133440000000000=2^{16}\cdot 5^{10}\cdot 29^{8}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $158.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{29} a^{8} - \frac{4}{29} a^{6} + \frac{6}{29} a^{4}$, $\frac{1}{29} a^{9} - \frac{4}{29} a^{7} + \frac{6}{29} a^{5}$, $\frac{1}{29} a^{10} - \frac{10}{29} a^{6} - \frac{5}{29} a^{4}$, $\frac{1}{29} a^{11} - \frac{10}{29} a^{7} - \frac{5}{29} a^{5}$, $\frac{1}{748490} a^{12} + \frac{12843}{748490} a^{10} - \frac{1647}{748490} a^{8} + \frac{5117}{25810} a^{6} - \frac{6268}{12905} a^{4} - \frac{1}{2}$, $\frac{1}{748490} a^{13} + \frac{12843}{748490} a^{11} - \frac{1647}{748490} a^{9} + \frac{5117}{25810} a^{7} - \frac{6268}{12905} a^{5} - \frac{1}{2} a$, $\frac{1}{65055382352498672360099846244472510} a^{14} + \frac{2542060005798856219761047147}{32527691176249336180049923122236255} a^{12} + \frac{213257305675970921971760098847828}{32527691176249336180049923122236255} a^{10} + \frac{7072516270631855456471871730892}{1121644523318942626898273211111595} a^{8} - \frac{271935393211137173889381049848069}{2243289046637885253796546422223190} a^{6} - \frac{18200065657352802610010755637327}{38677397355825607824078386590055} a^{4} + \frac{36302511327179977093299406887}{173830999352025203703723085798} a^{2} - \frac{632656086425366183975701235}{5994172391449144955300796062}$, $\frac{1}{65055382352498672360099846244472510} a^{15} + \frac{2542060005798856219761047147}{32527691176249336180049923122236255} a^{13} + \frac{213257305675970921971760098847828}{32527691176249336180049923122236255} a^{11} + \frac{7072516270631855456471871730892}{1121644523318942626898273211111595} a^{9} - \frac{271935393211137173889381049848069}{2243289046637885253796546422223190} a^{7} - \frac{18200065657352802610010755637327}{38677397355825607824078386590055} a^{5} + \frac{36302511327179977093299406887}{173830999352025203703723085798} a^{3} - \frac{632656086425366183975701235}{5994172391449144955300796062} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 108309651577 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

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 $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.2, 4.4.2225.1, 4.4.725.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$