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Properties

Label 15.5.41175521478...4944.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{15}\cdot 3^{5}\cdot 11^{13}\cdot 41^{13}\cdot 61^{13}$
Root discriminant $20{,}306.85$
Ramified primes $2, 3, 11, 41, 61$
Class number Not computed
Class group Not computed
Galois group $S_3 \times C_5$ (as 15T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5751424233849962588258304, 505655319767204772863856, 35886037118520058343136, 3008732613506684865072, 17888510070027150336, -5825118141734624712, -97851943474667568, 4866594417860856, 60044615624016, -2149890958601, -9448930490, 398483171, 474556, -32683, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 32683*x^13 + 474556*x^12 + 398483171*x^11 - 9448930490*x^10 - 2149890958601*x^9 + 60044615624016*x^8 + 4866594417860856*x^7 - 97851943474667568*x^6 - 5825118141734624712*x^5 + 17888510070027150336*x^4 + 3008732613506684865072*x^3 + 35886037118520058343136*x^2 + 505655319767204772863856*x + 5751424233849962588258304)
 
gp: K = bnfinit(x^15 - 2*x^14 - 32683*x^13 + 474556*x^12 + 398483171*x^11 - 9448930490*x^10 - 2149890958601*x^9 + 60044615624016*x^8 + 4866594417860856*x^7 - 97851943474667568*x^6 - 5825118141734624712*x^5 + 17888510070027150336*x^4 + 3008732613506684865072*x^3 + 35886037118520058343136*x^2 + 505655319767204772863856*x + 5751424233849962588258304, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 32683 x^{13} + 474556 x^{12} + 398483171 x^{11} - 9448930490 x^{10} - 2149890958601 x^{9} + 60044615624016 x^{8} + 4866594417860856 x^{7} - 97851943474667568 x^{6} - 5825118141734624712 x^{5} + 17888510070027150336 x^{4} + 3008732613506684865072 x^{3} + 35886037118520058343136 x^{2} + 505655319767204772863856 x + 5751424233849962588258304 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-41175521478048478091458686420635252885332273019437320881692114944=-\,2^{15}\cdot 3^{5}\cdot 11^{13}\cdot 41^{13}\cdot 61^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20{,}306.85$
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:  $2, 3, 11, 41, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} + \frac{1}{12} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{7} - \frac{1}{18} a^{5} + \frac{1}{6} a^{4} + \frac{7}{36} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{72} a^{8} - \frac{1}{36} a^{6} - \frac{17}{72} a^{4} - \frac{1}{6} a^{3} + \frac{5}{12} a^{2}$, $\frac{1}{432} a^{9} - \frac{1}{216} a^{7} - \frac{1}{72} a^{6} - \frac{23}{432} a^{5} - \frac{1}{72} a^{4} + \frac{1}{4} a^{3} + \frac{1}{12} a^{2} + \frac{5}{12} a$, $\frac{1}{1568787264} a^{10} - \frac{716041}{1568787264} a^{9} - \frac{788077}{196098408} a^{8} - \frac{2017343}{196098408} a^{7} + \frac{42890359}{1568787264} a^{6} + \frac{9344959}{142617024} a^{5} + \frac{373279}{14525808} a^{4} + \frac{1648685}{32683068} a^{3} + \frac{608461}{14525808} a^{2} - \frac{8265529}{43577424} a + \frac{104939}{907863}$, $\frac{1}{4706361792} a^{11} - \frac{1}{4706361792} a^{10} - \frac{44645}{73536903} a^{9} - \frac{14171}{4861944} a^{8} - \frac{9438041}{4706361792} a^{7} - \frac{48420787}{4706361792} a^{6} + \frac{27098545}{392196816} a^{5} + \frac{71221}{2391444} a^{4} + \frac{60180643}{130732272} a^{3} - \frac{29540569}{130732272} a^{2} - \frac{89042}{907863} a + \frac{413455}{907863}$, $\frac{1}{5558213276352} a^{12} - \frac{191}{2779106638176} a^{11} - \frac{677}{5558213276352} a^{10} - \frac{1274654165}{2779106638176} a^{9} + \frac{30063127015}{5558213276352} a^{8} - \frac{36904710419}{2779106638176} a^{7} - \frac{50497771777}{1852737758784} a^{6} - \frac{60206186519}{926368879392} a^{5} - \frac{7985197951}{154394813232} a^{4} - \frac{80632631}{19299351654} a^{3} - \frac{16818192275}{51464937744} a^{2} - \frac{999736525}{8577489624} a + \frac{137327315}{357395401}$, $\frac{1}{400191355897344} a^{13} + \frac{1}{18190516177152} a^{12} - \frac{41629}{400191355897344} a^{11} - \frac{1385}{6252989935896} a^{10} + \frac{29894142367}{36381032354304} a^{9} + \frac{850574371127}{200095677948672} a^{8} - \frac{213068867929}{36381032354304} a^{7} - \frac{7367120791}{336861410688} a^{6} - \frac{628141052837}{22232853105408} a^{5} + \frac{1260957070465}{5558213276352} a^{4} - \frac{787316826443}{3705475517568} a^{3} - \frac{430295759255}{926368879392} a^{2} - \frac{757418485535}{1852737758784} a + \frac{656522575}{9649675827}$, $\frac{1}{190817275728358900217250037614303831733700553663271203826848749857135837889729604559876301236835192151552} a^{14} + \frac{78357060467454609313763673862624924990336303225496130066781303791552794425456267721399959}{95408637864179450108625018807151915866850276831635601913424374928567918944864802279938150618417596075776} a^{13} + \frac{81197590360406886129009626232144945338964264715450506569187090713154513944508760993770421}{1042717353706879236159836271116414381058472970837547561895348359875059223441145380108613667960848044544} a^{12} + \frac{603677705203935000050071242775935142663767748844782136130082100924807628227672144378554555991}{7950719822014954175718751567262659655570856402636300159452031244047326578738733523328179218201466339648} a^{11} + \frac{11028159850751418134888921061342394899211861097237620468034457378093774329184843455329523517983}{63605758576119633405750012538101277244566851221090401275616249952378612629909868186625433745611730717184} a^{10} + \frac{11004918743628084005384491842226968835292150203236057781640689738808342603314752275459317479509854751}{10600959762686605567625002089683546207427808536848400212602708325396435438318311364437572290935288452864} a^{9} - \frac{470263526797228367434737217672302676676426523779764172115022400627386509938659600570274791619356985779}{190817275728358900217250037614303831733700553663271203826848749857135837889729604559876301236835192151552} a^{8} + \frac{59166839364648094387855398375117744995959398050775697126816946367643862483885448875224364115955259527}{47704318932089725054312509403575957933425138415817800956712187464283959472432401139969075309208798037888} a^{7} - \frac{76946031401871968297946065265050755877560701231832371683000640264426485114935427640030131815766055923}{3533653254228868522541667363227848735809269512282800070867569441798811812772770454812524096978429484288} a^{6} - \frac{6475950144259912160244595556759613822759494356078867790436839624836516641295436469783106446130933111}{120465451848711426904829569200949388720770551555095456961394412788595857253617174595881503306082823328} a^{5} + \frac{111291587781972231498008242134565007914743132113622955936707950541797790613532623476879910159576443143}{5300479881343302783812501044841773103713904268424200106301354162698217719159155682218786145467644226432} a^{4} - \frac{4788970971675868769238366979612092155731799403702689779874438964975414789597213828976502500466655065}{10773333092161184519944107814719051023808748513057317289230394639630523819429178215891841759080577696} a^{3} - \frac{11191985190266067833429736150777459476982854764821105505369597694977680817254244924519113981958815137}{32719011613230264097608031140998599405641384372988889545070087424063072340488615322338186083133606336} a^{2} + \frac{15274244833315641969213095647522449807249772422891561175019385967934541036964409416666188955207389745}{220853328389304282658854210201740545988079344517675004429223090112425738298298153425782756061151842768} a - \frac{351405695694851381627960148027528356437759307967266949931460765561008723251238963904197256633174147}{4601111008110505888726129379202928041418319677451562592275481044008869547881211529703807417940663391}$

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:  $9$
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

$C_5\times S_3$ (as 15T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.1.660264.1, 5.5.572829674183924641.1

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

Sibling fields

Galois closure: 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 $15$ $15$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ $15$ $15$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.5.4.4$x^{5} + 8856$$5$$1$$4$$C_5$$[\ ]_{5}$
41.10.9.5$x^{10} - 68864256$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$61$61.5.4.2$x^{5} + 122$$5$$1$$4$$C_5$$[\ ]_{5}$
61.10.9.4$x^{10} - 3904$$10$$1$$9$$C_{10}$$[\ ]_{10}$