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Properties

Label 16.6.55013835857...3056.2
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}$
Root discriminant $11{,}124.47$
Ramified primes $2, 3, 7, 294337$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-180141748647795219, 0, -142053998213105748, 0, -27550249139175866, 0, -288635660521604, 0, -93193540402, 0, 2572006372, 0, 1344530, 0, -5228, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5228*x^14 + 1344530*x^12 + 2572006372*x^10 - 93193540402*x^8 - 288635660521604*x^6 - 27550249139175866*x^4 - 142053998213105748*x^2 - 180141748647795219)
 
gp: K = bnfinit(x^16 - 5228*x^14 + 1344530*x^12 + 2572006372*x^10 - 93193540402*x^8 - 288635660521604*x^6 - 27550249139175866*x^4 - 142053998213105748*x^2 - 180141748647795219, 1)
 

Normalized defining polynomial

\( x^{16} - 5228 x^{14} + 1344530 x^{12} + 2572006372 x^{10} - 93193540402 x^{8} - 288635660521604 x^{6} - 27550249139175866 x^{4} - 142053998213105748 x^{2} - 180141748647795219 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-55013835857676939099088270698740000784693073140191887871554093056=-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}124.47$
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:  $2, 3, 7, 294337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{10} + \frac{1}{10} a^{8} + \frac{1}{5} a^{6} + \frac{3}{10} a^{4} + \frac{3}{10}$, $\frac{1}{27710} a^{13} - \frac{3266}{13855} a^{11} + \frac{5811}{27710} a^{9} - \frac{4874}{13855} a^{7} + \frac{11953}{27710} a^{5} + \frac{1183}{2771} a^{3} - \frac{12077}{27710} a$, $\frac{1}{738361347709983369418725370379675962569377596121557935485139730} a^{14} + \frac{14971861780691139350620569952169108919803732779869337346059991}{738361347709983369418725370379675962569377596121557935485139730} a^{12} - \frac{11072829093364138208789487389485466475200380949939354611230458}{73836134770998336941872537037967596256937759612155793548513973} a^{10} - \frac{36356709294173574819930610421141437135388933847915552895042895}{147672269541996673883745074075935192513875519224311587097027946} a^{8} - \frac{333255392205751974027477587479194061490602023927754967945040251}{738361347709983369418725370379675962569377596121557935485139730} a^{6} - \frac{169013620719910700046399598578515437643177207336421405403403091}{738361347709983369418725370379675962569377596121557935485139730} a^{4} - \frac{83010043577739203646787595798232335391369212580294752669543011}{369180673854991684709362685189837981284688798060778967742569865} a^{2} + \frac{89450662325280885647223041056601904215539695373058759853}{1634725448964147446092029787876795740656133078845886151010}$, $\frac{1}{120352899676727289215252235371887181898808548167813943484077775990} a^{15} + \frac{57116342489194391236906709468809797345761554362130370653076}{3539791166962567329860359863879034761729663181406292455414052235} a^{13} - \frac{27599486935971786941212088221735849765349596785107276920732594881}{120352899676727289215252235371887181898808548167813943484077775990} a^{11} + \frac{26364478554725710678178284970015348597993854551319022833672868053}{120352899676727289215252235371887181898808548167813943484077775990} a^{9} + \frac{6434403423042985530758905273000037545195081865065907346611916349}{24070579935345457843050447074377436379761709633562788696815555198} a^{7} - \frac{15197026277165173784684699365383860729175408284285627388116664431}{60176449838363644607626117685943590949404274083906971742038887995} a^{5} + \frac{57969077789892329826057242940944492735047374069469133826829121543}{120352899676727289215252235371887181898808548167813943484077775990} a^{3} - \frac{203473910040574314683937743701787953506614913265932124419153}{4529824219079652573121014542206600997358144761481950524448710} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $10$
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:  \( 80464149496200000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T994:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n994
Character table for t16n994 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), 4.4.2076841872.1, 8.6.975018691690245188654137344.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
294337Data not computed