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Properties

Label 16.8.28572634404...5609.1
Degree $16$
Signature $[8, 4]$
Discriminant $71^{12}\cdot 89^{15}$
Root discriminant $1644.36$
Ramified primes $71, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1333629299617792, 1561832763993728, -2797368182141824, 844893035355224, -98143453715880, -15136486766876, 4828308042143, -460309412308, 37964908341, 688001144, -61310410, 1651648, -527806, -6172, -37, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 37*x^14 - 6172*x^13 - 527806*x^12 + 1651648*x^11 - 61310410*x^10 + 688001144*x^9 + 37964908341*x^8 - 460309412308*x^7 + 4828308042143*x^6 - 15136486766876*x^5 - 98143453715880*x^4 + 844893035355224*x^3 - 2797368182141824*x^2 + 1561832763993728*x + 1333629299617792)
 
gp: K = bnfinit(x^16 - 4*x^15 - 37*x^14 - 6172*x^13 - 527806*x^12 + 1651648*x^11 - 61310410*x^10 + 688001144*x^9 + 37964908341*x^8 - 460309412308*x^7 + 4828308042143*x^6 - 15136486766876*x^5 - 98143453715880*x^4 + 844893035355224*x^3 - 2797368182141824*x^2 + 1561832763993728*x + 1333629299617792, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 37 x^{14} - 6172 x^{13} - 527806 x^{12} + 1651648 x^{11} - 61310410 x^{10} + 688001144 x^{9} + 37964908341 x^{8} - 460309412308 x^{7} + 4828308042143 x^{6} - 15136486766876 x^{5} - 98143453715880 x^{4} + 844893035355224 x^{3} - 2797368182141824 x^{2} + 1561832763993728 x + 1333629299617792 \)

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:  \(2857263440495560367103411585720766893299521110665609=71^{12}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1644.36$
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:  $71, 89$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4}$, $\frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4544} a^{8} + \frac{69}{4544} a^{7} - \frac{269}{4544} a^{6} + \frac{139}{4544} a^{5} - \frac{73}{1136} a^{4} - \frac{29}{568} a^{3} + \frac{19}{568} a^{2} + \frac{26}{71} a - \frac{35}{71}$, $\frac{1}{18176} a^{9} - \frac{1}{9088} a^{8} + \frac{57}{4544} a^{7} - \frac{463}{9088} a^{6} - \frac{221}{18176} a^{5} + \frac{255}{2272} a^{4} + \frac{161}{2272} a^{3} + \frac{563}{2272} a^{2} + \frac{143}{568} a$, $\frac{1}{18176} a^{10} - \frac{11}{9088} a^{7} - \frac{889}{18176} a^{6} - \frac{1}{9088} a^{5} - \frac{241}{2272} a^{4} + \frac{565}{2272} a^{3} + \frac{141}{1136} a^{2} + \frac{141}{284} a - \frac{28}{71}$, $\frac{1}{1599488} a^{11} + \frac{3}{1599488} a^{10} + \frac{15}{799744} a^{9} - \frac{41}{799744} a^{8} - \frac{14563}{1599488} a^{7} + \frac{94511}{1599488} a^{6} + \frac{21147}{399872} a^{5} + \frac{2889}{24992} a^{4} + \frac{44295}{199936} a^{3} - \frac{127}{24992} a^{2} - \frac{973}{12496} a + \frac{25}{781}$, $\frac{1}{25591808} a^{12} - \frac{1}{3198976} a^{11} - \frac{443}{25591808} a^{10} + \frac{7}{6397952} a^{9} - \frac{461}{25591808} a^{8} - \frac{8227}{799744} a^{7} + \frac{1039311}{25591808} a^{6} - \frac{121075}{6397952} a^{5} + \frac{269135}{3198976} a^{4} - \frac{174629}{3198976} a^{3} + \frac{5429}{199936} a^{2} - \frac{39827}{199936} a + \frac{245}{568}$, $\frac{1}{102367232} a^{13} + \frac{1}{102367232} a^{12} - \frac{3}{102367232} a^{11} - \frac{2423}{102367232} a^{10} - \frac{1745}{102367232} a^{9} - \frac{479}{9306112} a^{8} + \frac{1216111}{102367232} a^{7} - \frac{1415045}{102367232} a^{6} + \frac{19097}{2326528} a^{5} - \frac{747183}{6397952} a^{4} + \frac{637443}{12795904} a^{3} - \frac{48523}{399872} a^{2} - \frac{59339}{799744} a - \frac{4721}{24992}$, $\frac{1}{3259493050744832} a^{14} + \frac{6305559}{1629746525372416} a^{13} + \frac{19659733}{1629746525372416} a^{12} - \frac{25003869}{148158775033856} a^{11} - \frac{18750921807}{814873262686208} a^{10} - \frac{33230301097}{1629746525372416} a^{9} + \frac{139528833375}{1629746525372416} a^{8} - \frac{23540037221121}{1629746525372416} a^{7} + \frac{23023539630699}{3259493050744832} a^{6} + \frac{23806930950299}{814873262686208} a^{5} - \frac{26600315717379}{407436631343104} a^{4} + \frac{78524157982119}{407436631343104} a^{3} + \frac{5231985628247}{25464789458944} a^{2} - \frac{10275084257359}{25464789458944} a + \frac{307984538019}{795774670592}$, $\frac{1}{675587816537036499851461153855354816695209629026467967708222818615296} a^{15} + \frac{79132349486070941207126956507463328694241055493841299}{675587816537036499851461153855354816695209629026467967708222818615296} a^{14} + \frac{38806473167614278178285276621674425051886118983900827771849}{42224238533564781240716322115959676043450601814154247981763926163456} a^{13} - \frac{1124258875234994047092670691708951018563301340268891337606603}{168896954134259124962865288463838704173802407256616991927055704653824} a^{12} - \frac{28038869386321315884563381592706259256795230364308345640668761}{337793908268518249925730576927677408347604814513233983854111409307648} a^{11} - \frac{7174584017494784531383838881773457495021172630971090907915813279}{337793908268518249925730576927677408347604814513233983854111409307648} a^{10} - \frac{109260045958186374176981300547264088241504552213163209619964759}{168896954134259124962865288463838704173802407256616991927055704653824} a^{9} + \frac{2415722728599776430262429375448073051377419306930963460198517453}{168896954134259124962865288463838704173802407256616991927055704653824} a^{8} - \frac{167393082713162461318080924293310619999197174090013229104405311775}{675587816537036499851461153855354816695209629026467967708222818615296} a^{7} - \frac{1364863399470444988054449641814555023699434021005279230938193893917}{675587816537036499851461153855354816695209629026467967708222818615296} a^{6} - \frac{9534428230352044676104710373743037534567992388740861684678415553439}{168896954134259124962865288463838704173802407256616991927055704653824} a^{5} + \frac{22001667982014631215534152632045853030703565362060711700401575709}{959641784853745028198098229908174455532968223048960181403725594624} a^{4} - \frac{26284156332357014071401535552733690775551909925836159980410099937}{108128651814506482050489941398104164003714729357629316214504292352} a^{3} - \frac{104015446959370312109797114776375644013823584058849340133456485791}{1319507454173899413772385066123739876357831306692320249430122692608} a^{2} + \frac{644193635614974225415906680059972380967569563491223598136887904757}{5278029816695597655089540264494959505431325226769280997720490770432} a + \frac{33414323209729158941621020176527214414764150460213687112579613327}{164938431771737426721548133265467484544728913336540031178765336576}$

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:  \( 58327836957700000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.1123992572569351274249.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$71$71.8.6.3$x^{8} - 71 x^{4} + 55451$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
71.8.6.3$x^{8} - 71 x^{4} + 55451$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
89Data not computed