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Properties

Label 16.8.84097996886...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 29^{2}\cdot 61^{2}$
Root discriminant $41.72$
Ramified primes $2, 3, 5, 29, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15359, -16890, 62442, 9004, -27147, 34596, -39830, 3690, 7981, -4086, 1650, -688, 315, -90, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 90*x^13 + 315*x^12 - 688*x^11 + 1650*x^10 - 4086*x^9 + 7981*x^8 + 3690*x^7 - 39830*x^6 + 34596*x^5 - 27147*x^4 + 9004*x^3 + 62442*x^2 - 16890*x - 15359)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 90*x^13 + 315*x^12 - 688*x^11 + 1650*x^10 - 4086*x^9 + 7981*x^8 + 3690*x^7 - 39830*x^6 + 34596*x^5 - 27147*x^4 + 9004*x^3 + 62442*x^2 - 16890*x - 15359, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 90 x^{13} + 315 x^{12} - 688 x^{11} + 1650 x^{10} - 4086 x^{9} + 7981 x^{8} + 3690 x^{7} - 39830 x^{6} + 34596 x^{5} - 27147 x^{4} + 9004 x^{3} + 62442 x^{2} - 16890 x - 15359 \)

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:  \(84097996886016000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 29^{2}\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.72$
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:  $2, 3, 5, 29, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41196637636731115577673706934732296783471} a^{15} - \frac{2467778084031596260340154400263185915439}{41196637636731115577673706934732296783471} a^{14} + \frac{4243253441978809378894521318571261546147}{41196637636731115577673706934732296783471} a^{13} - \frac{12702019010239162196711622042943457553441}{41196637636731115577673706934732296783471} a^{12} - \frac{5781483570701609687578593583855850460790}{41196637636731115577673706934732296783471} a^{11} + \frac{3550593209775955531184992614963846438877}{41196637636731115577673706934732296783471} a^{10} + \frac{20459348051617573012933418426861537636717}{41196637636731115577673706934732296783471} a^{9} + \frac{7662055089067548117450348323622306895533}{41196637636731115577673706934732296783471} a^{8} - \frac{10033289742537415626682139380740122270591}{41196637636731115577673706934732296783471} a^{7} - \frac{11501425661795501473047527223697537913478}{41196637636731115577673706934732296783471} a^{6} + \frac{20015846005934480964674677333059905708014}{41196637636731115577673706934732296783471} a^{5} + \frac{2981932991064188276702756074594489980038}{41196637636731115577673706934732296783471} a^{4} + \frac{7153998406295249818849045988992345174632}{41196637636731115577673706934732296783471} a^{3} + \frac{11439191299973718966239875970371367537685}{41196637636731115577673706934732296783471} a^{2} + \frac{3380004000152224897872730295505601331209}{41196637636731115577673706934732296783471} a - \frac{995424207247648076446708077924228975740}{41196637636731115577673706934732296783471}$

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:  \( 6981573.43489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
3Data not computed
5Data not computed
$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}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$