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Properties

Label 15.15.2029833056...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{5}\cdot 61^{11}\cdot 397^{3}\cdot 3449539^{10}$
Root discriminant $2{,}633{,}281.90$
Ramified primes $5, 61, 397, 3449539$
Class number Not computed
Class group Not computed
Galois group 15T70

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-42062243539779429694928280813246936486645620471338796875, -143899055360198090303599834796122108858596671245045000, -160246170978138178880960112966590413992426168835000, -78383192783129061050530216704529630578004400600, -14342265880481731614912529111681563352984845, 1741434416253074466115924536784589820944, 1045698931780078435682650479628885030, 72017999183738288707629373330605, -23053765882121416072139784690, -3218288153233940153496245, 219030836843089675149, 49200196895337165, -779807896785, -350703120, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 350703120*x^13 - 779807896785*x^12 + 49200196895337165*x^11 + 219030836843089675149*x^10 - 3218288153233940153496245*x^9 - 23053765882121416072139784690*x^8 + 72017999183738288707629373330605*x^7 + 1045698931780078435682650479628885030*x^6 + 1741434416253074466115924536784589820944*x^5 - 14342265880481731614912529111681563352984845*x^4 - 78383192783129061050530216704529630578004400600*x^3 - 160246170978138178880960112966590413992426168835000*x^2 - 143899055360198090303599834796122108858596671245045000*x - 42062243539779429694928280813246936486645620471338796875)
 
gp: K = bnfinit(x^15 - 5*x^14 - 350703120*x^13 - 779807896785*x^12 + 49200196895337165*x^11 + 219030836843089675149*x^10 - 3218288153233940153496245*x^9 - 23053765882121416072139784690*x^8 + 72017999183738288707629373330605*x^7 + 1045698931780078435682650479628885030*x^6 + 1741434416253074466115924536784589820944*x^5 - 14342265880481731614912529111681563352984845*x^4 - 78383192783129061050530216704529630578004400600*x^3 - 160246170978138178880960112966590413992426168835000*x^2 - 143899055360198090303599834796122108858596671245045000*x - 42062243539779429694928280813246936486645620471338796875, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 350703120 x^{13} - 779807896785 x^{12} + 49200196895337165 x^{11} + 219030836843089675149 x^{10} - 3218288153233940153496245 x^{9} - 23053765882121416072139784690 x^{8} + 72017999183738288707629373330605 x^{7} + 1045698931780078435682650479628885030 x^{6} + 1741434416253074466115924536784589820944 x^{5} - 14342265880481731614912529111681563352984845 x^{4} - 78383192783129061050530216704529630578004400600 x^{3} - 160246170978138178880960112966590413992426168835000 x^{2} - 143899055360198090303599834796122108858596671245045000 x - 42062243539779429694928280813246936486645620471338796875 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2029833056749933990877437692446176993120127718812297756397657810230365055931909199614663014853125=5^{5}\cdot 61^{11}\cdot 397^{3}\cdot 3449539^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2{,}633{,}281.90$
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:  $5, 61, 397, 3449539$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{1052109395} a^{3} - \frac{1}{1052109395} a^{2} - \frac{70140626}{1052109395} a - \frac{63905904}{210421879}$, $\frac{1}{1052109395} a^{4} - \frac{70140627}{1052109395} a^{2} - \frac{389670146}{1052109395} a - \frac{63905904}{210421879}$, $\frac{1}{1052109395} a^{5} - \frac{459810773}{1052109395} a^{2} - \frac{85727432}{1052109395} a - \frac{42603936}{210421879}$, $\frac{1}{1106934179047266025} a^{6} - \frac{2}{1106934179047266025} a^{5} - \frac{140281251}{1106934179047266025} a^{4} - \frac{498777788}{1106934179047266025} a^{3} + \frac{4919708054730916}{1106934179047266025} a^{2} + \frac{8964800223311808}{221386835809453205} a + \frac{4083964566057216}{44277367161890641}$, $\frac{1}{1106934179047266025} a^{7} - \frac{28056251}{221386835809453205} a^{5} + \frac{54553821}{221386835809453205} a^{4} + \frac{77934029}{221386835809453205} a^{3} - \frac{14212488744364598}{1106934179047266025} a^{2} - \frac{29214559091818104}{221386835809453205} a - \frac{6583904802418899}{44277367161890641}$, $\frac{1}{1106934179047266025} a^{8} - \frac{1558681}{221386835809453205} a^{5} + \frac{54553821}{221386835809453205} a^{4} + \frac{275366902}{1106934179047266025} a^{3} - \frac{81254131488913727}{221386835809453205} a^{2} + \frac{5199098290828359}{221386835809453205} a + \frac{10097750180112231}{44277367161890641}$, $\frac{1}{1164615849422240733966804875} a^{9} - \frac{3}{1164615849422240733966804875} a^{8} - \frac{1683375}{9316926795377925871734439} a^{7} + \frac{102872918}{232923169884448146793360975} a^{6} + \frac{2951824369910416}{232923169884448146793360975} a^{5} - \frac{27878341975380213}{1164615849422240733966804875} a^{4} - \frac{477576046739169156}{1164615849422240733966804875} a^{3} + \frac{23004855658504979471332}{232923169884448146793360975} a^{2} - \frac{17602218026805649567105702}{46584633976889629358672195} a - \frac{2921464093543147674191579}{9316926795377925871734439}$, $\frac{1}{1164615849422240733966804875} a^{10} - \frac{210421884}{1164615849422240733966804875} a^{8} - \frac{23380207}{232923169884448146793360975} a^{7} + \frac{46760416}{232923169884448146793360975} a^{6} + \frac{45917269890963567}{1164615849422240733966804875} a^{5} + \frac{11989509278977203}{46584633976889629358672195} a^{4} + \frac{46828326856160902}{1164615849422240733966804875} a^{3} - \frac{47545721677100919080031133}{232923169884448146793360975} a^{2} + \frac{1644826909774420589586769}{9316926795377925871734439} a + \frac{3747162589519619337457965}{9316926795377925871734439}$, $\frac{1}{1164615849422240733966804875} a^{11} + \frac{303942708}{1164615849422240733966804875} a^{8} + \frac{9352084}{46584633976889629358672195} a^{7} + \frac{62347212}{1164615849422240733966804875} a^{6} + \frac{14744545424271787}{46584633976889629358672195} a^{5} - \frac{1195306689999028}{9316926795377925871734439} a^{4} + \frac{145647634970135841}{1164615849422240733966804875} a^{3} - \frac{34654442499641681026044472}{232923169884448146793360975} a^{2} + \frac{10401615315789291772521734}{46584633976889629358672195} a + \frac{1702132422468775542426720}{9316926795377925871734439}$, $\frac{1}{1560231234648548734263954411624389932230212605773125} a^{12} - \frac{4}{1560231234648548734263954411624389932230212605773125} a^{11} - \frac{280562498}{1560231234648548734263954411624389932230212605773125} a^{10} + \frac{547160911189345551945863}{1560231234648548734263954411624389932230212605773125} a^{9} - \frac{328296540809958095686264}{312046246929709746852790882324877986446042521154625} a^{8} - \frac{115134624859170452990383501015633}{1560231234648548734263954411624389932230212605773125} a^{7} - \frac{398005763366098544603415749856448}{1560231234648548734263954411624389932230212605773125} a^{6} + \frac{8075615918447403532795200419403069765204}{1560231234648548734263954411624389932230212605773125} a^{5} + \frac{80059590244851962155160555934029498591911}{1560231234648548734263954411624389932230212605773125} a^{4} - \frac{147370630780916952248417421334021793368661}{312046246929709746852790882324877986446042521154625} a^{3} - \frac{131189526300091817011268469437271613993180230639}{62409249385941949370558176464975597289208504230925} a^{2} + \frac{540690309070225849905169351991733930221604421338}{12481849877188389874111635292995119457841700846185} a - \frac{830615027597912569994141348511104066615320426337}{2496369975437677974822327058599023891568340169237}$, $\frac{1}{1560231234648548734263954411624389932230212605773125} a^{13} - \frac{280562514}{1560231234648548734263954411624389932230212605773125} a^{11} + \frac{547160911189344429695871}{1560231234648548734263954411624389932230212605773125} a^{10} + \frac{547160940707591729352132}{1560231234648548734263954411624389932230212605773125} a^{9} - \frac{115134631425101269189545414740913}{1560231234648548734263954411624389932230212605773125} a^{8} + \frac{110192504210527415373814153189989}{312046246929709746852790882324877986446042521154625} a^{7} + \frac{254070959979458479179408250133662}{1560231234648548734263954411624389932230212605773125} a^{6} - \frac{69214098506903339646776914958659816436648}{1560231234648548734263954411624389932230212605773125} a^{5} - \frac{539922688992268234504727634413889336939911}{1560231234648548734263954411624389932230212605773125} a^{4} - \frac{113465165420123659629614644921652640852639}{312046246929709746852790882324877986446042521154625} a^{3} - \frac{140672326624323965721596227928195041079803167272}{62409249385941949370558176464975597289208504230925} a^{2} + \frac{1795518605971727577194123694813603856161831938684}{12481849877188389874111635292995119457841700846185} a - \frac{466725028451341519085867589769012774243077799083}{2496369975437677974822327058599023891568340169237}$, $\frac{1}{332597623053678938800207235545870666612113673329138895363392760270625} a^{14} + \frac{345781533433127}{66519524610735787760041447109174133322422734665827779072678552054125} a^{13} + \frac{12713080396481257}{66519524610735787760041447109174133322422734665827779072678552054125} a^{12} - \frac{5164985405504795584222702038699085882852}{66519524610735787760041447109174133322422734665827779072678552054125} a^{11} + \frac{8375820154583083415247027161643073476697}{66519524610735787760041447109174133322422734665827779072678552054125} a^{10} - \frac{118902031147441019582245564624886680379581}{332597623053678938800207235545870666612113673329138895363392760270625} a^{9} - \frac{1707017394311058770491540353178106132990500408231}{66519524610735787760041447109174133322422734665827779072678552054125} a^{8} + \frac{1590918249257392973028133483870497131292648032674}{66519524610735787760041447109174133322422734665827779072678552054125} a^{7} + \frac{2715077147049034953224538296117019259791141671066}{66519524610735787760041447109174133322422734665827779072678552054125} a^{6} - \frac{23955745484398354814291184340519975658750153320887125746576}{66519524610735787760041447109174133322422734665827779072678552054125} a^{5} - \frac{126244282551714306158319919688426489223404765434129144157841}{332597623053678938800207235545870666612113673329138895363392760270625} a^{4} + \frac{798287968526040460383874964295053778915195701494340303537}{2660780984429431510401657884366965332896909386633111162907142082165} a^{3} + \frac{1315708959712191243829350956003308815560105103212044932217423996358}{13303904922147157552008289421834826664484546933165555814535710410825} a^{2} - \frac{1006666877285532698557677800127553100988025348400783502075381186776}{2660780984429431510401657884366965332896909386633111162907142082165} a + \frac{97618108807332677126177107205072254928128778706773252636986322392}{532156196885886302080331576873393066579381877326622232581428416433}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T70:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 19440
The 36 conjugacy class representatives for [3^4:2]S(5)
Character table for [3^4:2]S(5) is not computed

Intermediate fields

5.5.24217.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61Data not computed
397Data not computed
3449539Data not computed