↑

Properties

Label 15.15.1421441829...2192.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 37^{7}\cdot 109964946338501^{4}$
Root discriminant $47{,}516.96$
Ramified primes $2, 37, 109964946338501$
Class number Not computed
Class group Not computed
Galois group 15T31

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2179454923921583313202432, -1831737201071142890547968, -331068493589414852154368, -12434446258906106164736, 755175919354707053120, 60370334006953788288, 860907007423227232, -28319323358595040, -987072530657200, -6650961625168, 93277045460, 1231831520, -2007357, -61709, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 61709*x^13 - 2007357*x^12 + 1231831520*x^11 + 93277045460*x^10 - 6650961625168*x^9 - 987072530657200*x^8 - 28319323358595040*x^7 + 860907007423227232*x^6 + 60370334006953788288*x^5 + 755175919354707053120*x^4 - 12434446258906106164736*x^3 - 331068493589414852154368*x^2 - 1831737201071142890547968*x - 2179454923921583313202432)
 
gp: K = bnfinit(x^15 - 7*x^14 - 61709*x^13 - 2007357*x^12 + 1231831520*x^11 + 93277045460*x^10 - 6650961625168*x^9 - 987072530657200*x^8 - 28319323358595040*x^7 + 860907007423227232*x^6 + 60370334006953788288*x^5 + 755175919354707053120*x^4 - 12434446258906106164736*x^3 - 331068493589414852154368*x^2 - 1831737201071142890547968*x - 2179454923921583313202432, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 61709 x^{13} - 2007357 x^{12} + 1231831520 x^{11} + 93277045460 x^{10} - 6650961625168 x^{9} - 987072530657200 x^{8} - 28319323358595040 x^{7} + 860907007423227232 x^{6} + 60370334006953788288 x^{5} + 755175919354707053120 x^{4} - 12434446258906106164736 x^{3} - 331068493589414852154368 x^{2} - 1831737201071142890547968 x - 2179454923921583313202432 \)

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:  \(14214418299374473825888879195675870012903878163752261874979816918152192=2^{10}\cdot 37^{7}\cdot 109964946338501^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47{,}516.96$
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:  $2, 37, 109964946338501$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{9} + \frac{3}{64} a^{8} + \frac{1}{64} a^{7} - \frac{3}{64} a^{6} - \frac{3}{32} a^{5} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{64} a^{10} + \frac{1}{32} a^{7} - \frac{5}{64} a^{6} + \frac{5}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{9} + \frac{7}{128} a^{8} + \frac{5}{64} a^{6} - \frac{1}{32} a^{5} - \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{512} a^{12} - \frac{1}{512} a^{11} + \frac{1}{512} a^{10} + \frac{1}{512} a^{9} + \frac{15}{256} a^{8} - \frac{1}{64} a^{7} + \frac{5}{128} a^{6} + \frac{1}{8} a^{5} + \frac{3}{32} a^{4} + \frac{3}{32} a^{3} - \frac{5}{16} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{18944} a^{13} - \frac{3}{4736} a^{12} - \frac{1}{4736} a^{11} + \frac{31}{9472} a^{10} + \frac{3}{18944} a^{9} + \frac{79}{9472} a^{8} - \frac{289}{4736} a^{7} - \frac{269}{4736} a^{6} - \frac{109}{592} a^{5} + \frac{23}{592} a^{4} - \frac{531}{1184} a^{3} + \frac{125}{592} a^{2} - \frac{29}{296} a + \frac{97}{296}$, $\frac{1}{2482552279302292081204121649747881603403679903123892221637027051413641221500086256915604630195107075353784562911539997268887715924986211814400} a^{14} - \frac{7634946181991605495072159314421872263009347835886274230099568286841011242051893831299208115357715745242365792891041959315861431787124153}{496510455860458416240824329949576320680735980624778444327405410282728244300017251383120926039021415070756912582307999453777543184997242362880} a^{13} - \frac{1619553881961120754922040849066845143587715872823024318781079214030727879014668082379803151969809690282833508615207959622703085548952925489}{2482552279302292081204121649747881603403679903123892221637027051413641221500086256915604630195107075353784562911539997268887715924986211814400} a^{12} + \frac{1696447635884875227707052565021371622724848398846050841658103434833352673899210365240507712787314533636778062931512395506226631608832208751}{496510455860458416240824329949576320680735980624778444327405410282728244300017251383120926039021415070756912582307999453777543184997242362880} a^{11} - \frac{367788012534228202299064312516373062705816187624375622923066673473997703610180955279775333303896708160708294720441032125633069352551622721}{124127613965114604060206082487394080170183995156194611081851352570682061075004312845780231509755353767689228145576999863444385796249310590720} a^{10} + \frac{909211372396834775534437976056759299422866303218518307176116836810632585919190522056446923846261133973372500181519947814221422189671009297}{248255227930229208120412164974788160340367990312389222163702705141364122150008625691560463019510707535378456291153999726888771592498621181440} a^{9} + \frac{482393447322775226467912363233698278547982652185127042144941224465019947569386606151476332870303174038348728497725641659093791741181240059}{310319034912786510150515206218485200425459987890486527704628381426705152687510782114450578774388384419223070363942499658610964490623276476800} a^{8} - \frac{2594874759701419643576172043283941357148422540935131311258073820218640028545670378360651332416069730486194799602376289848285023928781247543}{77579758728196627537628801554621300106364996972621631926157095356676288171877695528612644693597096104805767590985624914652741122655819119200} a^{7} - \frac{34922183362266459155147234418365739658207743463612937116845874780437996905935566467355182923004689374304726435682705101251798672481183811229}{310319034912786510150515206218485200425459987890486527704628381426705152687510782114450578774388384419223070363942499658610964490623276476800} a^{6} - \frac{6018840590616383775372878238342441035174481549444679553118787186967571842129672426370051454263206217466768787895864958369123874589234058457}{155159517456393255075257603109242600212729993945243263852314190713352576343755391057225289387194192209611535181971249829305482245311638238400} a^{5} + \frac{7397877043969143420293101139332428175966823833415453782780110808051841388193075650535858618044563987948260807055635756779893180394138322031}{38789879364098313768814400777310650053182498486310815963078547678338144085938847764306322346798548052402883795492812457326370561327909559600} a^{4} - \frac{7616207800926934658239171351388690417139118313671717323834578865441657722181162805733078838619095434719458308555841560196084978969599098161}{77579758728196627537628801554621300106364996972621631926157095356676288171877695528612644693597096104805767590985624914652741122655819119200} a^{3} - \frac{119847931400585836121739502452877180671715330821519906027562803379399064862619945237774759287055979573543737688060939622639479832507694021}{484873492051228922110180009716383125664781231078885199538481845979226801074235597053829029334981850655036047443660155716579632016598869495} a^{2} + \frac{560909776524897171029161652170682861078553211399630463038998847613321426290505815648060396804452323831111943006691504612634636930276713683}{2424367460256144610550900048581915628323906155394425997692409229896134005371177985269145146674909253275180237218300778582898160082994347475} a - \frac{8417790898500394924014439913512634512442204791940864881966053246667393076238645319839795780623134803445961367837625041226615953519724747393}{19394939682049156884407200388655325026591249243155407981539273839169072042969423882153161173399274026201441897746406228663185280663954779800}$

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

15T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 750
The 35 conjugacy class representatives for 1/2[5^3:2]S(3)
Character table for 1/2[5^3:2]S(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 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 $15$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
37Data not computed
109964946338501Data not computed