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Properties

Label 16.8.75677711862...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{6}$
Root discriminant $85.10$
Ramified primes $2, 5, 29, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T868

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6661561, 0, -7859145, 0, -5053047, 0, 814900, 0, 157399, 0, -8910, 0, -938, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 20*x^14 - 938*x^12 - 8910*x^10 + 157399*x^8 + 814900*x^6 - 5053047*x^4 - 7859145*x^2 + 6661561)
 
gp: K = bnfinit(x^16 + 20*x^14 - 938*x^12 - 8910*x^10 + 157399*x^8 + 814900*x^6 - 5053047*x^4 - 7859145*x^2 + 6661561, 1)
 

Normalized defining polynomial

\( x^{16} + 20 x^{14} - 938 x^{12} - 8910 x^{10} + 157399 x^{8} + 814900 x^{6} - 5053047 x^{4} - 7859145 x^{2} + 6661561 \)

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:  \(7567771186285806117913600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15} a^{8} - \frac{1}{3} a^{6} + \frac{2}{5} a^{4} - \frac{1}{15}$, $\frac{1}{15} a^{9} - \frac{1}{3} a^{7} + \frac{2}{5} a^{5} - \frac{1}{15} a$, $\frac{1}{15} a^{10} - \frac{4}{15} a^{6} - \frac{1}{15} a^{2} - \frac{1}{3}$, $\frac{1}{15} a^{11} - \frac{4}{15} a^{7} - \frac{1}{15} a^{3} - \frac{1}{3} a$, $\frac{1}{435} a^{12} - \frac{3}{145} a^{10} - \frac{2}{87} a^{8} + \frac{17}{145} a^{6} - \frac{187}{435} a^{4} - \frac{4}{15} a^{2} - \frac{2}{5}$, $\frac{1}{435} a^{13} - \frac{3}{145} a^{11} - \frac{2}{87} a^{9} + \frac{17}{145} a^{7} - \frac{187}{435} a^{5} - \frac{4}{15} a^{3} - \frac{2}{5} a$, $\frac{1}{1126196963002277581889235} a^{14} - \frac{1191905623475291394358}{1126196963002277581889235} a^{12} - \frac{6047436135143824022501}{225239392600455516377847} a^{10} + \frac{3596577188871506385476}{375398987667425860629745} a^{8} + \frac{1388009530605591756763}{1126196963002277581889235} a^{6} - \frac{306493794510440809820461}{1126196963002277581889235} a^{4} + \frac{5378424196966103775274}{38834378034561295927215} a^{2} + \frac{169666306211035705939}{436341326231025796935}$, $\frac{1}{1126196963002277581889235} a^{15} - \frac{1191905623475291394358}{1126196963002277581889235} a^{13} - \frac{6047436135143824022501}{225239392600455516377847} a^{11} + \frac{3596577188871506385476}{375398987667425860629745} a^{9} + \frac{1388009530605591756763}{1126196963002277581889235} a^{7} - \frac{306493794510440809820461}{1126196963002277581889235} a^{5} + \frac{5378424196966103775274}{38834378034561295927215} a^{3} + \frac{169666306211035705939}{436341326231025796935} a$

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:  \( 1454447486.28 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T868:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n868 are not computed
Character table for t16n868 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.64525.2, 4.4.2225.1, 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 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$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$