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Properties

Label 16.12.2837991076...0656.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{66}\cdot 17^{2}\cdot 191^{4}$
Root discriminant $92.43$
Ramified primes $2, 17, 191$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![693889, 0, -1126216, 0, -153564, 0, 394376, 0, -90786, 0, 5160, 0, 324, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 324*x^12 + 5160*x^10 - 90786*x^8 + 394376*x^6 - 153564*x^4 - 1126216*x^2 + 693889)
 
gp: K = bnfinit(x^16 - 40*x^14 + 324*x^12 + 5160*x^10 - 90786*x^8 + 394376*x^6 - 153564*x^4 - 1126216*x^2 + 693889, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} + 324 x^{12} + 5160 x^{10} - 90786 x^{8} + 394376 x^{6} - 153564 x^{4} - 1126216 x^{2} + 693889 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28379910764965177995302489030656=2^{66}\cdot 17^{2}\cdot 191^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.43$
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:  $2, 17, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{170} a^{10} - \frac{12}{85} a^{8} - \frac{11}{85} a^{6} + \frac{27}{85} a^{4} + \frac{1}{34} a^{2} - \frac{2}{5}$, $\frac{1}{340} a^{11} - \frac{1}{340} a^{10} + \frac{61}{340} a^{9} - \frac{61}{340} a^{8} - \frac{11}{170} a^{7} + \frac{11}{170} a^{6} + \frac{27}{170} a^{5} - \frac{27}{170} a^{4} + \frac{1}{68} a^{3} - \frac{1}{68} a^{2} + \frac{1}{20} a - \frac{1}{20}$, $\frac{1}{69700} a^{12} - \frac{14}{17425} a^{10} - \frac{2569}{69700} a^{8} - \frac{7101}{34850} a^{6} - \frac{363}{69700} a^{4} - \frac{5469}{34850} a^{2} - \frac{797}{4100}$, $\frac{1}{487900} a^{13} + \frac{149}{487900} a^{11} - \frac{1}{340} a^{10} - \frac{14941}{121975} a^{9} - \frac{61}{340} a^{8} + \frac{30172}{121975} a^{7} + \frac{11}{170} a^{6} + \frac{10707}{487900} a^{5} - \frac{27}{170} a^{4} - \frac{219013}{487900} a^{3} - \frac{1}{68} a^{2} - \frac{148}{7175} a - \frac{1}{20}$, $\frac{1}{14828259239500} a^{14} - \frac{60583493}{14828259239500} a^{12} + \frac{1582552359}{872250543500} a^{10} + \frac{3302578925801}{14828259239500} a^{8} + \frac{2823350462811}{14828259239500} a^{6} + \frac{3634072871593}{14828259239500} a^{4} + \frac{219960156321}{872250543500} a^{2} - \frac{42181117}{1047119500}$, $\frac{1}{103797814676500} a^{15} - \frac{60583493}{103797814676500} a^{13} + \frac{6713437909}{6105753804500} a^{11} + \frac{8623307241151}{103797814676500} a^{9} + \frac{8318528886861}{103797814676500} a^{7} - \frac{43554681531757}{103797814676500} a^{5} + \frac{2426240942821}{6105753804500} a^{3} - \frac{3602387417}{7329836500} a$

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:  $13$
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:  \( 36468100129.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.195584.1, \(\Q(\zeta_{16})^+\), 4.4.391168.1, 8.8.2448198467584.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
191Data not computed