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Properties

Label 16.8.22947318992...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{12}\cdot 29^{4}\cdot 89^{2}$
Root discriminant $38.46$
Ramified primes $2, 5, 29, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1061, 2494, -1728, -5214, 3328, 6666, -3232, -6234, -406, 2080, 970, 132, 7, 10, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 + 10*x^13 + 7*x^12 + 132*x^11 + 970*x^10 + 2080*x^9 - 406*x^8 - 6234*x^7 - 3232*x^6 + 6666*x^5 + 3328*x^4 - 5214*x^3 - 1728*x^2 + 2494*x + 1061)
 
gp: K = bnfinit(x^16 - 4*x^15 - 10*x^14 + 10*x^13 + 7*x^12 + 132*x^11 + 970*x^10 + 2080*x^9 - 406*x^8 - 6234*x^7 - 3232*x^6 + 6666*x^5 + 3328*x^4 - 5214*x^3 - 1728*x^2 + 2494*x + 1061, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 10 x^{14} + 10 x^{13} + 7 x^{12} + 132 x^{11} + 970 x^{10} + 2080 x^{9} - 406 x^{8} - 6234 x^{7} - 3232 x^{6} + 6666 x^{5} + 3328 x^{4} - 5214 x^{3} - 1728 x^{2} + 2494 x + 1061 \)

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:  \(22947318992896000000000000=2^{24}\cdot 5^{12}\cdot 29^{4}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{84776100712433629813297373281} a^{15} + \frac{41738810305532790605041257122}{84776100712433629813297373281} a^{14} + \frac{37034679036548873936485569311}{84776100712433629813297373281} a^{13} - \frac{17685476566569973278161862425}{84776100712433629813297373281} a^{12} - \frac{10461353320436581188343454222}{84776100712433629813297373281} a^{11} - \frac{37821945841867850413924475671}{84776100712433629813297373281} a^{10} - \frac{38442600389698287941619814784}{84776100712433629813297373281} a^{9} + \frac{28678059732332377504944591733}{84776100712433629813297373281} a^{8} + \frac{1622276948722902310209643553}{84776100712433629813297373281} a^{7} - \frac{12674380441699678619515492006}{84776100712433629813297373281} a^{6} + \frac{9445390170214797460057290408}{84776100712433629813297373281} a^{5} + \frac{14565637746803540815273601545}{84776100712433629813297373281} a^{4} + \frac{697698464994006544646997867}{84776100712433629813297373281} a^{3} + \frac{34493399250155215747698747935}{84776100712433629813297373281} a^{2} + \frac{34412573751096739861856661386}{84776100712433629813297373281} a + \frac{11880336024488556447476990564}{84776100712433629813297373281}$

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:  $11$
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:  \( 4247487.37425 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{20})^+\), 4.4.58000.1, 8.8.3364000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
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}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$