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Properties

Label 16.8.32684701122...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{12}\cdot 761$
Root discriminant $34.05$
Ramified primes $2, 5, 761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1379

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 812, 476, -5740, 8818, -5296, 512, 1204, -1563, 2072, -1612, 660, -120, -12, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 12*x^13 - 120*x^12 + 660*x^11 - 1612*x^10 + 2072*x^9 - 1563*x^8 + 1204*x^7 + 512*x^6 - 5296*x^5 + 8818*x^4 - 5740*x^3 + 476*x^2 + 812*x - 199)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 12*x^13 - 120*x^12 + 660*x^11 - 1612*x^10 + 2072*x^9 - 1563*x^8 + 1204*x^7 + 512*x^6 - 5296*x^5 + 8818*x^4 - 5740*x^3 + 476*x^2 + 812*x - 199, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 12 x^{13} - 120 x^{12} + 660 x^{11} - 1612 x^{10} + 2072 x^{9} - 1563 x^{8} + 1204 x^{7} + 512 x^{6} - 5296 x^{5} + 8818 x^{4} - 5740 x^{3} + 476 x^{2} + 812 x - 199 \)

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:  \(3268470112256000000000000=2^{44}\cdot 5^{12}\cdot 761\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.05$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{2665} a^{14} - \frac{216}{2665} a^{13} - \frac{258}{2665} a^{12} + \frac{288}{2665} a^{11} - \frac{22}{205} a^{10} + \frac{11}{41} a^{9} - \frac{7}{205} a^{8} + \frac{235}{533} a^{7} + \frac{868}{2665} a^{6} + \frac{189}{533} a^{5} - \frac{851}{2665} a^{4} - \frac{1233}{2665} a^{3} - \frac{187}{2665} a^{2} - \frac{966}{2665} a - \frac{994}{2665}$, $\frac{1}{1592771540956944105146675} a^{15} - \frac{159678736205881694517}{1592771540956944105146675} a^{14} + \frac{70644121247740591683728}{1592771540956944105146675} a^{13} + \frac{54833696733172251880911}{1592771540956944105146675} a^{12} - \frac{379042219985387047242204}{1592771540956944105146675} a^{11} - \frac{51222336202768922903973}{122520887765918777318975} a^{10} - \frac{18536012288053283348197}{122520887765918777318975} a^{9} + \frac{71575128768634065102806}{1592771540956944105146675} a^{8} - \frac{230113937478361918263042}{1592771540956944105146675} a^{7} - \frac{187071163613867888510648}{1592771540956944105146675} a^{6} - \frac{369352743784103501773606}{1592771540956944105146675} a^{5} + \frac{671039233071090760270848}{1592771540956944105146675} a^{4} - \frac{549513750156236711108774}{1592771540956944105146675} a^{3} - \frac{2216571143295886699208}{11138262524174434301725} a^{2} - \frac{11767236875013059347058}{1592771540956944105146675} a - \frac{684452373388580682015991}{1592771540956944105146675}$

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:  \( 2021338.38887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1379:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1379 are not computed
Character table for t16n1379 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
761Data not computed