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Properties

Label 15.7.11091565920...8749.1
Degree $15$
Signature $[7, 4]$
Discriminant $229^{7}\cdot 541^{4}\cdot 250229701^{4}$
Root discriminant $11{,}739.95$
Ramified primes $229, 541, 250229701$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 15T48

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2664795024960878546944, 1061857604265640723456, 172778346922476550912, 15819541345795830848, 927255109184758544, 31236364450391152, 35582772872952, -57828570766468, -2380181235483, -27717468995, 1106306238, 38842907, 215385, -15691, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 15691*x^13 + 215385*x^12 + 38842907*x^11 + 1106306238*x^10 - 27717468995*x^9 - 2380181235483*x^8 - 57828570766468*x^7 + 35582772872952*x^6 + 31236364450391152*x^5 + 927255109184758544*x^4 + 15819541345795830848*x^3 + 172778346922476550912*x^2 + 1061857604265640723456*x + 2664795024960878546944)
 
gp: K = bnfinit(x^15 - 3*x^14 - 15691*x^13 + 215385*x^12 + 38842907*x^11 + 1106306238*x^10 - 27717468995*x^9 - 2380181235483*x^8 - 57828570766468*x^7 + 35582772872952*x^6 + 31236364450391152*x^5 + 927255109184758544*x^4 + 15819541345795830848*x^3 + 172778346922476550912*x^2 + 1061857604265640723456*x + 2664795024960878546944, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 15691 x^{13} + 215385 x^{12} + 38842907 x^{11} + 1106306238 x^{10} - 27717468995 x^{9} - 2380181235483 x^{8} - 57828570766468 x^{7} + 35582772872952 x^{6} + 31236364450391152 x^{5} + 927255109184758544 x^{4} + 15819541345795830848 x^{3} + 172778346922476550912 x^{2} + 1061857604265640723456 x + 2664795024960878546944 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11091565920936809225054175428820300456860415694668000344708749=229^{7}\cdot 541^{4}\cdot 250229701^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}739.95$
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:  $229, 541, 250229701$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{3}{32} a^{7} - \frac{3}{32} a^{6} + \frac{1}{8} a^{5} + \frac{9}{32} a^{4} - \frac{1}{32} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{3}{64} a^{8} - \frac{3}{64} a^{7} + \frac{1}{16} a^{6} - \frac{23}{64} a^{5} + \frac{31}{64} a^{4} - \frac{13}{32} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{256} a^{12} + \frac{1}{256} a^{11} - \frac{3}{256} a^{10} + \frac{1}{256} a^{9} + \frac{3}{256} a^{8} - \frac{1}{128} a^{7} + \frac{49}{256} a^{6} - \frac{79}{256} a^{5} + \frac{9}{64} a^{4} + \frac{31}{64} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{234496} a^{13} - \frac{223}{234496} a^{12} - \frac{715}{234496} a^{11} + \frac{553}{234496} a^{10} + \frac{11883}{234496} a^{9} + \frac{3203}{117248} a^{8} + \frac{86153}{234496} a^{7} + \frac{19281}{234496} a^{6} + \frac{29223}{58624} a^{5} - \frac{3487}{58624} a^{4} + \frac{4657}{14656} a^{3} + \frac{7}{3664} a^{2} - \frac{63}{229} a + \frac{78}{229}$, $\frac{1}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{14} - \frac{156736955700840781822083352349735074296492655646955207794844593308296578640651435766906415826843277501048276790490824061249}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{13} - \frac{205987824756731062424882800886125953554696510133738028105415889707903569323244434012413401758674109271505324404135283718542093}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{12} + \frac{271604096890179204307644302196326377838390380910646878779787172271242149849713209872814067973587980909423894541676975165280767}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{11} - \frac{1698532535513170920203340924502311878410417203488227278397246571954662417061577321259713865926312687250349397995453516100784935}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{10} + \frac{282284824561168665307150048377762157904089862167291207781211003599686949318777973775128741350216589716975011216614126581491179}{8324240319594528745466396216460242671837894575492499874022027749862363398377809628772493185524137616580055377531746495956777472} a^{9} - \frac{27449729608748366437897723256789060465349609263026520250265132542545181589055576075614691241104712322050482518108108672186829667}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{8} - \frac{52506368094363600130355559135234656812062729097126615739680337267397647745010059667263478387740120069252624490749708137269349153}{133187845113512459927462339463363882749406313207879997984352443997797814374044954060359890968386201865280886040507943935308439552} a^{7} - \frac{20994283423517595038008999012979318605718954663232325395899115788589517128298983865424315521330543659171147282399906329164453507}{66593922556756229963731169731681941374703156603939998992176221998898907187022477030179945484193100932640443020253971967654219776} a^{6} + \frac{1110020689367138806123449446660390157510030037231270367240924757391526088646205222097541485676635461833986446239377076338320443}{33296961278378114981865584865840970687351578301969999496088110999449453593511238515089972742096550466320221510126985983827109888} a^{5} + \frac{2829282060262259270294314515875068954496325943254260471031564528902645886293065470259233481868084498842844785456904769489214377}{16648480639189057490932792432920485343675789150984999748044055499724726796755619257544986371048275233160110755063492991913554944} a^{4} + \frac{1385300818209672652920594597466626692402288981009787723550553234543984919373823708716217068035917693475905049003783839614581109}{4162120159797264372733198108230121335918947287746249937011013874931181699188904814386246592762068808290027688765873247978388736} a^{3} - \frac{349234210212566725578478643676262444799051398699211476836626186369431942372745880201032966592368917906714667860909425084815741}{1040530039949316093183299527057530333979736821936562484252753468732795424797226203596561648190517202072506922191468311994597184} a^{2} + \frac{56064090984657282688607105844763226679779285045085775993709696098527062255537423229046403797112987332981165311943912518852641}{260132509987329023295824881764382583494934205484140621063188367183198856199306550899140412047629300518126730547867077998649296} a + \frac{28866997737049976549208952104891822896669737317078080405936529879270418761898868926656445688224866462030391279019971962907555}{65033127496832255823956220441095645873733551371035155265797091795799714049826637724785103011907325129531682636966769499662324}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $10$
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:  \( 1682212466290000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 26 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

3.3.229.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ $15$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $15$ $15$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
229Data not computed
541Data not computed
250229701Data not computed