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Properties

Label 16.0.79999954991...9664.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{60}\cdot 7^{4}\cdot 17^{2}$
Root discriminant $31.18$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 0, -816, 0, 1332, 0, -928, 0, 450, 0, -16, 0, 12, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^12 - 16*x^10 + 450*x^8 - 928*x^6 + 1332*x^4 - 816*x^2 + 289)
 
gp: K = bnfinit(x^16 + 12*x^12 - 16*x^10 + 450*x^8 - 928*x^6 + 1332*x^4 - 816*x^2 + 289, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{12} - 16 x^{10} + 450 x^{8} - 928 x^{6} + 1332 x^{4} - 816 x^{2} + 289 \)

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:  \(799999549910140441329664=2^{60}\cdot 7^{4}\cdot 17^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.18$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{8} a$, $\frac{1}{48} a^{10} - \frac{1}{16} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{16} a^{2} + \frac{11}{48}$, $\frac{1}{48} a^{11} - \frac{1}{16} a^{9} + \frac{1}{8} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{16} a^{3} + \frac{11}{48} a - \frac{1}{2}$, $\frac{1}{144} a^{12} - \frac{1}{144} a^{10} + \frac{1}{24} a^{8} - \frac{1}{24} a^{6} - \frac{1}{2} a^{5} - \frac{1}{16} a^{4} + \frac{41}{144} a^{2} - \frac{1}{2} a - \frac{2}{9}$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{11} + \frac{1}{24} a^{9} - \frac{1}{24} a^{7} - \frac{1}{2} a^{6} - \frac{1}{16} a^{5} + \frac{41}{144} a^{3} - \frac{1}{2} a^{2} - \frac{2}{9} a$, $\frac{1}{10700208} a^{14} - \frac{157}{157356} a^{12} + \frac{64357}{10700208} a^{10} - \frac{5471}{1783368} a^{8} - \frac{1}{2} a^{7} + \frac{229259}{3566736} a^{6} - \frac{1}{2} a^{5} + \frac{38695}{1337526} a^{4} - \frac{1}{2} a^{3} + \frac{834281}{10700208} a^{2} - \frac{1}{2} a + \frac{20369}{314712}$, $\frac{1}{10700208} a^{15} - \frac{157}{157356} a^{13} + \frac{64357}{10700208} a^{11} - \frac{5471}{1783368} a^{9} + \frac{229259}{3566736} a^{7} - \frac{1}{2} a^{6} + \frac{38695}{1337526} a^{5} - \frac{1}{2} a^{4} + \frac{834281}{10700208} a^{3} - \frac{1}{2} a^{2} + \frac{20369}{314712} a - \frac{1}{2}$

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:  $7$
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:  \( 212362.166213 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.14336.1, \(\Q(\zeta_{16})^+\), 4.0.7168.1, 8.0.3288334336.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$