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Properties

Label 15.15.1297349726...2757.1
Degree $15$
Signature $[15, 0]$
Discriminant $19^{5}\cdot 103^{5}\cdot 244301^{12}$
Root discriminant $255{,}586.04$
Ramified primes $19, 103, 244301$
Class number Not computed
Class group Not computed
Galois group 15T51

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1788115490230017965919208633168, -367428540321761276369592413252, 50654221628876906428539791586, -1524999325458746904578062859, 5737266459200156336848041, 334208520172135190753528, -3902925851520568641401, -8002301492179106409, 309436858997805328, -997902540485763, -5439011478175, 29992349079, 28250964, -293157, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 293157*x^13 + 28250964*x^12 + 29992349079*x^11 - 5439011478175*x^10 - 997902540485763*x^9 + 309436858997805328*x^8 - 8002301492179106409*x^7 - 3902925851520568641401*x^6 + 334208520172135190753528*x^5 + 5737266459200156336848041*x^4 - 1524999325458746904578062859*x^3 + 50654221628876906428539791586*x^2 - 367428540321761276369592413252*x - 1788115490230017965919208633168)
 
gp: K = bnfinit(x^15 - 3*x^14 - 293157*x^13 + 28250964*x^12 + 29992349079*x^11 - 5439011478175*x^10 - 997902540485763*x^9 + 309436858997805328*x^8 - 8002301492179106409*x^7 - 3902925851520568641401*x^6 + 334208520172135190753528*x^5 + 5737266459200156336848041*x^4 - 1524999325458746904578062859*x^3 + 50654221628876906428539791586*x^2 - 367428540321761276369592413252*x - 1788115490230017965919208633168, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 293157 x^{13} + 28250964 x^{12} + 29992349079 x^{11} - 5439011478175 x^{10} - 997902540485763 x^{9} + 309436858997805328 x^{8} - 8002301492179106409 x^{7} - 3902925851520568641401 x^{6} + 334208520172135190753528 x^{5} + 5737266459200156336848041 x^{4} - 1524999325458746904578062859 x^{3} + 50654221628876906428539791586 x^{2} - 367428540321761276369592413252 x - 1788115490230017965919208633168 \)

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:  \(1297349726010274931651495881999903609711923026220354873585739003189726789242882757=19^{5}\cdot 103^{5}\cdot 244301^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255{,}586.04$
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:  $19, 103, 244301$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{244301} a^{5} - \frac{1}{244301} a^{4} - \frac{97720}{244301} a^{3} + \frac{19544}{244301} a^{2} + \frac{95766}{244301} a - \frac{91779}{244301}$, $\frac{1}{244301} a^{6} - \frac{97721}{244301} a^{4} - \frac{78176}{244301} a^{3} + \frac{115310}{244301} a^{2} + \frac{3987}{244301} a - \frac{91779}{244301}$, $\frac{1}{244301} a^{7} + \frac{68404}{244301} a^{4} + \frac{56678}{244301} a^{3} - \frac{82007}{244301} a^{2} + \frac{63401}{244301} a + \frac{42653}{244301}$, $\frac{1}{488602} a^{8} - \frac{1}{488602} a^{7} - \frac{1}{488602} a^{5} - \frac{93811}{244301} a^{4} + \frac{39127}{244301} a^{3} + \frac{26580}{244301} a^{2} + \frac{37337}{488602} a + \frac{26372}{244301}$, $\frac{1}{488602} a^{9} - \frac{1}{488602} a^{7} - \frac{1}{488602} a^{6} - \frac{1}{488602} a^{5} + \frac{95806}{244301} a^{4} + \frac{5511}{244301} a^{3} + \frac{16855}{488602} a^{2} + \frac{48585}{488602} a + \frac{46746}{244301}$, $\frac{1}{119365957202} a^{10} - \frac{1}{59682978601} a^{9} + \frac{24431}{59682978601} a^{8} + \frac{117264}{59682978601} a^{7} - \frac{130945}{119365957202} a^{6} + \frac{80287}{119365957202} a^{5} - \frac{4870167484}{59682978601} a^{4} - \frac{49537794051}{119365957202} a^{3} - \frac{58854872663}{119365957202} a^{2} + \frac{16748362383}{119365957202} a - \frac{11837051022}{59682978601}$, $\frac{1}{119365957202} a^{11} + \frac{24429}{59682978601} a^{9} + \frac{87951}{119365957202} a^{8} + \frac{46905}{59682978601} a^{7} - \frac{181603}{119365957202} a^{6} - \frac{137825}{119365957202} a^{5} + \frac{53021611961}{119365957202} a^{4} - \frac{33732718385}{119365957202} a^{3} + \frac{11284665915}{119365957202} a^{2} - \frac{14919938257}{119365957202} a - \frac{28907518066}{59682978601}$, $\frac{1}{119365957202} a^{12} - \frac{29317}{59682978601} a^{9} + \frac{51793}{59682978601} a^{8} + \frac{93889}{59682978601} a^{7} + \frac{31349}{59682978601} a^{6} + \frac{33819}{59682978601} a^{5} + \frac{44670584809}{119365957202} a^{4} + \frac{16076940981}{119365957202} a^{3} + \frac{271269431}{59682978601} a^{2} + \frac{4857983347}{119365957202} a - \frac{19594066837}{59682978601}$, $\frac{1}{119365957202} a^{13} - \frac{6841}{59682978601} a^{9} + \frac{79}{59682978601} a^{8} + \frac{81381}{59682978601} a^{7} + \frac{65168}{59682978601} a^{6} + \frac{14647}{119365957202} a^{5} - \frac{37419092743}{119365957202} a^{4} + \frac{10939920761}{59682978601} a^{3} - \frac{31403373781}{119365957202} a^{2} - \frac{3121713515}{59682978601} a + \frac{13117720369}{59682978601}$, $\frac{1}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{14} - \frac{171927268992977933929970900867855179288178043165219540415573891324994219088277876436470112881557114911387467}{53378379674322815044984263720340356154645030082735885306256607111669551216154884618122424159086661784178602236731407748} a^{13} + \frac{581042098357781110072443426156505821884600606866018143905344309346898472259606444784284875459957792349024363}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{12} + \frac{207197622580459631387127405603163475004206522667088360647841634830781720755734892630546340856504209925643467}{66722974592903518806230329650425445193306287603419856632820758889586939020193605772653030198858327230223252795914259685} a^{11} + \frac{75947000774492902762529915934375228791433313475287086899615989808429160669001034347603851702137395701552003}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{10} + \frac{39984115768574872252197168702263347800047820895480612426618348055532827311892149383453997388803133158067680052389}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{9} + \frac{59161607337477092855444355588351499102593081721244911770899456829786311948035083106329435521250022093900062234619}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{8} + \frac{1165847315043767326889559086932652037198303290740781696253944508473220273735568868528299877259884914761950248955}{26689189837161407522492131860170178077322515041367942653128303555834775608077442309061212079543330892089301118365703874} a^{7} - \frac{95876481722454418432676186906993619754795305904490787656817247791106394947399988092307424535963587060694750166869}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{6} + \frac{46233262438298328715169788796076682859149394629782354580763950443980807310796817524352668936370921803613587031547}{24262899851964915929538301691063798252111377310334493321025730505304341461888583917328374617766664447353910107605185340} a^{5} - \frac{21134472287944197422510106421010847311816825220389617783751666681472828049979993971566288551008686366204105261375775354}{66722974592903518806230329650425445193306287603419856632820758889586939020193605772653030198858327230223252795914259685} a^{4} - \frac{22510691789206204043779276433640341364584359162209235878796841167119583602233085308442976907512917803919425965047512347}{266891898371614075224921318601701780773225150413679426531283035558347756080774423090612120795433308920893011183657038740} a^{3} + \frac{11178148935067011524458236543236903353995811267347750483645022018978132391811495324637995841532466624768424180653959569}{53378379674322815044984263720340356154645030082735885306256607111669551216154884618122424159086661784178602236731407748} a^{2} + \frac{27910700829096057186149652413216794288359500906244775434666174779940916506894612071573252229234812957399708860930180334}{66722974592903518806230329650425445193306287603419856632820758889586939020193605772653030198858327230223252795914259685} a - \frac{4005891992239978408716841688832700283116946925632191172668165316802770493697317489017995682758221859699815823629546261}{66722974592903518806230329650425445193306287603419856632820758889586939020193605772653030198858327230223252795914259685}$

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

15T51:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 38 conjugacy class representatives for [1/2.D(5)^3]S(3)
Character table for [1/2.D(5)^3]S(3) is not computed

Intermediate fields

3.3.1957.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R $15$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
103Data not computed
244301Data not computed