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Properties

Label 36.36.1092445132...1957.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{18}\cdot 13^{33}\cdot 19^{24}$
Root discriminant $129.47$
Ramified primes $3, 13, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1923169, 2030710, -85804113, 27696257, 1224100668, -1005405651, -8221358547, 8975560111, 29752270500, -39397313544, -60074993232, 98707302558, 62851529470, -147685422825, -18736332340, 131524497452, -26075281081, -67146393883, 29791639547, 18261358636, -13374176751, -2057742341, 3263481172, -160121713, -470447773, 82284019, 40364678, -11630750, -1892871, 881049, 28176, -38375, 1465, 906, -74, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 - 74*x^34 + 906*x^33 + 1465*x^32 - 38375*x^31 + 28176*x^30 + 881049*x^29 - 1892871*x^28 - 11630750*x^27 + 40364678*x^26 + 82284019*x^25 - 470447773*x^24 - 160121713*x^23 + 3263481172*x^22 - 2057742341*x^21 - 13374176751*x^20 + 18261358636*x^19 + 29791639547*x^18 - 67146393883*x^17 - 26075281081*x^16 + 131524497452*x^15 - 18736332340*x^14 - 147685422825*x^13 + 62851529470*x^12 + 98707302558*x^11 - 60074993232*x^10 - 39397313544*x^9 + 29752270500*x^8 + 8975560111*x^7 - 8221358547*x^6 - 1005405651*x^5 + 1224100668*x^4 + 27696257*x^3 - 85804113*x^2 + 2030710*x + 1923169)
 
gp: K = bnfinit(x^36 - 9*x^35 - 74*x^34 + 906*x^33 + 1465*x^32 - 38375*x^31 + 28176*x^30 + 881049*x^29 - 1892871*x^28 - 11630750*x^27 + 40364678*x^26 + 82284019*x^25 - 470447773*x^24 - 160121713*x^23 + 3263481172*x^22 - 2057742341*x^21 - 13374176751*x^20 + 18261358636*x^19 + 29791639547*x^18 - 67146393883*x^17 - 26075281081*x^16 + 131524497452*x^15 - 18736332340*x^14 - 147685422825*x^13 + 62851529470*x^12 + 98707302558*x^11 - 60074993232*x^10 - 39397313544*x^9 + 29752270500*x^8 + 8975560111*x^7 - 8221358547*x^6 - 1005405651*x^5 + 1224100668*x^4 + 27696257*x^3 - 85804113*x^2 + 2030710*x + 1923169, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} - 74 x^{34} + 906 x^{33} + 1465 x^{32} - 38375 x^{31} + 28176 x^{30} + 881049 x^{29} - 1892871 x^{28} - 11630750 x^{27} + 40364678 x^{26} + 82284019 x^{25} - 470447773 x^{24} - 160121713 x^{23} + 3263481172 x^{22} - 2057742341 x^{21} - 13374176751 x^{20} + 18261358636 x^{19} + 29791639547 x^{18} - 67146393883 x^{17} - 26075281081 x^{16} + 131524497452 x^{15} - 18736332340 x^{14} - 147685422825 x^{13} + 62851529470 x^{12} + 98707302558 x^{11} - 60074993232 x^{10} - 39397313544 x^{9} + 29752270500 x^{8} + 8975560111 x^{7} - 8221358547 x^{6} - 1005405651 x^{5} + 1224100668 x^{4} + 27696257 x^{3} - 85804113 x^{2} + 2030710 x + 1923169 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10924451324402501022100983006336490133035923719372600153527586154197811111957=3^{18}\cdot 13^{33}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.47$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(741=3\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{741}(1,·)$, $\chi_{741}(391,·)$, $\chi_{741}(64,·)$, $\chi_{741}(11,·)$, $\chi_{741}(400,·)$, $\chi_{741}(20,·)$, $\chi_{741}(277,·)$, $\chi_{741}(406,·)$, $\chi_{741}(410,·)$, $\chi_{741}(539,·)$, $\chi_{741}(305,·)$, $\chi_{741}(220,·)$, $\chi_{741}(172,·)$, $\chi_{741}(685,·)$, $\chi_{741}(49,·)$, $\chi_{741}(562,·)$, $\chi_{741}(695,·)$, $\chi_{741}(571,·)$, $\chi_{741}(704,·)$, $\chi_{741}(197,·)$, $\chi_{741}(710,·)$, $\chi_{741}(334,·)$, $\chi_{741}(83,·)$, $\chi_{741}(596,·)$, $\chi_{741}(590,·)$, $\chi_{741}(476,·)$, $\chi_{741}(353,·)$, $\chi_{741}(178,·)$, $\chi_{741}(362,·)$, $\chi_{741}(235,·)$, $\chi_{741}(239,·)$, $\chi_{741}(628,·)$, $\chi_{741}(121,·)$, $\chi_{741}(634,·)$, $\chi_{741}(125,·)$, $\chi_{741}(254,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{3} a^{30} + \frac{1}{3} a^{29} + \frac{1}{3} a^{27} - \frac{1}{3} a^{26} - \frac{1}{3} a^{23} + \frac{1}{3} a^{22} + \frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{19} - \frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{31} - \frac{1}{3} a^{29} + \frac{1}{3} a^{28} + \frac{1}{3} a^{27} + \frac{1}{3} a^{26} - \frac{1}{3} a^{24} - \frac{1}{3} a^{23} + \frac{1}{3} a^{21} - \frac{1}{3} a^{20} - \frac{1}{3} a^{19} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{32} - \frac{1}{3} a^{29} + \frac{1}{3} a^{28} - \frac{1}{3} a^{27} - \frac{1}{3} a^{26} - \frac{1}{3} a^{25} - \frac{1}{3} a^{24} - \frac{1}{3} a^{23} - \frac{1}{3} a^{22} + \frac{1}{3} a^{20} + \frac{1}{3} a^{19} - \frac{1}{3} a^{18} + \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{33} - \frac{1}{3} a^{29} - \frac{1}{3} a^{28} + \frac{1}{3} a^{26} - \frac{1}{3} a^{25} - \frac{1}{3} a^{24} + \frac{1}{3} a^{23} + \frac{1}{3} a^{22} - \frac{1}{3} a^{21} + \frac{1}{3} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{933} a^{34} + \frac{32}{933} a^{33} + \frac{121}{933} a^{32} - \frac{7}{311} a^{31} + \frac{110}{933} a^{30} - \frac{145}{933} a^{29} + \frac{125}{933} a^{28} + \frac{23}{311} a^{27} - \frac{27}{311} a^{26} - \frac{439}{933} a^{25} - \frac{308}{933} a^{24} - \frac{322}{933} a^{23} - \frac{3}{311} a^{22} - \frac{464}{933} a^{21} + \frac{232}{933} a^{20} + \frac{77}{933} a^{19} + \frac{350}{933} a^{18} + \frac{173}{933} a^{17} - \frac{19}{933} a^{16} - \frac{139}{311} a^{15} + \frac{230}{933} a^{14} - \frac{3}{311} a^{13} + \frac{117}{311} a^{12} + \frac{434}{933} a^{11} + \frac{191}{933} a^{10} + \frac{421}{933} a^{9} + \frac{214}{933} a^{8} + \frac{112}{933} a^{7} - \frac{126}{311} a^{6} + \frac{250}{933} a^{5} + \frac{35}{933} a^{4} - \frac{278}{933} a^{3} - \frac{36}{311} a^{2} - \frac{85}{933} a + \frac{23}{311}$, $\frac{1}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{35} - \frac{53971546158531615565938607071878980111033311370050871968355603638976688763130241462086082530280792143361879017508811109709755763356894829218893373487}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{34} - \frac{41697136687855404613484278672642817784772050768964463999655335875486149607287939778881379842294599134493803928009780283964110750198859038810948309306}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{33} - \frac{4698054185935017486933260977225204749477530842911903021675147523453160857249505657462170124882520609182606142738702172117362678702457943356480259536727}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{32} - \frac{5119026147763410244307149454147471081763118230762091427596726789673590112013947101902137289355909494216767831572427485854939721117057401779238809450052}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{31} + \frac{5603602231352122106966289323114083769911610341395817645529872017519120828691093105784099553607470992994179610757053533319643274162055031731445318244078}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{30} - \frac{31897957669013585978655580773814117368151747678649865968268641904945221604571284295440599615687072960571761603787518395470806159425315416832605838003729}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{29} + \frac{2082107805577520275402100359395541364966919872002688908904209858827639900050392959014782576432446728590339787075109473715677706475985722170807460900816}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{28} - \frac{47187828920750607147536097819801780398415832048738697172381602181052795131461790320231766588455222181329368176392117560533215630803409988228502137975961}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{27} - \frac{28056772710914156629476121082357152086863649994604120035914685068288310037183842797403176629791768736829536132301793234227401092427484205932347499307720}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{26} - \frac{35458292988583278439912895810987183756807027402575786670978169946604327557958830204638747585878456194688590012395489395360555143078523107900063567439053}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{25} + \frac{8561914857540200316503753600814829870336700624433392481503570681847045740887965180934507370184413166634837129363850284383357692524391119509486070439114}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{24} + \frac{26052801463425945439276952555505609209203581691856151324779767122580029144778228836454771026104286864490134642920432867068702731782679893914266575697503}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{23} - \frac{159128455382868842418583243154475259289856321987863605128211549254842947019764407529990071956915283410841549558815301145460156365594491631992760650881}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{22} - \frac{2700208706745555563830938519402785346589249556126800221589046212952585745565351793733938659240081391162006544934762895186878419189550245343562476282225}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{21} - \frac{26064385044235521049418934820659307324580207663059569366894684027204047822169634502248733742338062300773752498819745638974839253105888937306119436509240}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{20} + \frac{43985759119609067546829968289846315811177719724113879086785673023959621735514064226840644332440971748938411531740374554683626652803517753284291451301044}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{19} + \frac{12216786711786466249137538159405706610147267272497201749204329496253834360021472303482943966343159848276073791649758567955163759123573483516543596435227}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{18} + \frac{20520943921800966137498793041583458901456271909853900279472909253185431217833359362858830298837640274272191934565522335285906058725210920026834440615598}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{17} + \frac{4847619231662113694488195200997268043927536369238584525050085563911714369536818588777831806664019340238043303950652503129372191615127296300473911878484}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{16} + \frac{13459161171867652515186152327903321029466841099239469959803100249361415657619417750233438977024526072093277263182030711650543607006115923460194370446568}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{15} + \frac{32892776693124949510537230335843359710888719591680466343419316029649184590820327732584245279621059238197403448663636337054996076898326678401836067465159}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{14} - \frac{12373501587391187396753204104213587931536447698155206438995378909748967156064157830038787096488349573394266652125884525124491869704248041737267076642994}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{13} + \frac{44117132865529447434086533346827006706254213616032770885025331955551100790583136996843488632416505214395791352281208573138014124658338335773056772274581}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{12} + \frac{8960952970934121587050493198666029982150978660293243886307625721375359490809453890511840465872127767448025546001060874970072061562156430950308410189840}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{11} - \frac{39036917458336495878193894090366123553060897749541652437884753101705177897621419422879508034623019581417611199685713112308007503354119964879795945768720}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{10} + \frac{10434257859809483440402089339972966102621194150398916343764142509154969915663710767449760443823375812464483446449978428057260702904412333978179554841291}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{9} - \frac{50152723802887605097736656242525411267668372926670394219523010702830658128170431441030051597536073874326581076404145111950704777757221373604052727714625}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{8} + \frac{15095483409145551120962981181739142753067516257194739006902372150795865722130388603117511720196988689275383423424147570144382584444311808473178643592267}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{7} + \frac{13554360138943218055716772597903845030063946556488317060916449538515025850277487649513799296224453101581612856137997324885560721929255445635839653163361}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{6} - \frac{51316719375726598436866050389566579970359079580687568473974651662255259968841794657908833608710081928703310819746974289195752494191044627354930776600821}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{5} - \frac{7882151693796368155808858122940684351502012795744863660680496569217456346624549598104678847984948786489446637234156627224017149457400070819419088185311}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{4} + \frac{33888494008887368508186259097743146567473300665018895517253156527804678048711809206410884281346844950093436042369253549160764273180857690708259187390575}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a^{3} - \frac{13964868447369154526868996662147104537512738284029692268976570327082919172261538737521593588451262339966246414244092773373972562704765624852046319368005}{34894422333743675130730194567757227106463206120237266649152353303204694302937385864516586567713180686720758048198478548887758067182300532685173268884699} a^{2} - \frac{9832225557262796096011480862761653572247341209434501615959393801402888361829132469434532601376405801996475113374980479780217656423844528516290986116702}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097} a + \frac{11903485072521395804312010576077666370940512707326853118835733238160758289385639166295839078466423241498880570451746716833765678714812488257556806394474}{104683267001231025392190583703271681319389618360711799947457059909614082908812157593549759703139542060162274144595435646663274201546901598055519806654097}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $35$
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:  \( 98353320439532980000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.61009.2, 3.3.61009.1, 3.3.361.1, 4.4.19773.1, \(\Q(\zeta_{13})^+\), 6.6.48387275053.2, 6.6.48387275053.1, 6.6.286315237.1, 9.9.227081481823729.1, \(\Q(\zeta_{39})^+\), 12.12.22188769124117165900170893.2, 12.12.22188769124117165900170893.1, 12.12.131294491858681455030597.1, 18.18.113290500653811459555808941573877.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed
19Data not computed