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Properties

Label 36.36.4286755119...2533.1
Degree $36$
Signature $[36, 0]$
Discriminant $11^{18}\cdot 37^{35}$
Root discriminant $111.00$
Ramified primes $11, 37$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1324838279, 15659396372, -15659396372, -256697207395, 256697207395, 1286656880618, -1286656880618, -2828954020750, 2828954020750, 3458784856340, -3458784856340, -2676523987366, 2676523987366, 1413681908438, -1413681908438, -534035184802, 534035184802, 148620561113, -148620561113, -31025687812, 31025687812, 4903561973, -4903561973, -587601367, 587601367, 53034356, -53034356, -3546007, 3546007, 170273, -170273, -5551, 5551, 110, -110, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279)
 
gp: K = bnfinit(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 110 x^{34} + 110 x^{33} + 5551 x^{32} - 5551 x^{31} - 170273 x^{30} + 170273 x^{29} + 3546007 x^{28} - 3546007 x^{27} - 53034356 x^{26} + 53034356 x^{25} + 587601367 x^{24} - 587601367 x^{23} - 4903561973 x^{22} + 4903561973 x^{21} + 31025687812 x^{20} - 31025687812 x^{19} - 148620561113 x^{18} + 148620561113 x^{17} + 534035184802 x^{16} - 534035184802 x^{15} - 1413681908438 x^{14} + 1413681908438 x^{13} + 2676523987366 x^{12} - 2676523987366 x^{11} - 3458784856340 x^{10} + 3458784856340 x^{9} + 2828954020750 x^{8} - 2828954020750 x^{7} - 1286656880618 x^{6} + 1286656880618 x^{5} + 256697207395 x^{4} - 256697207395 x^{3} - 15659396372 x^{2} + 15659396372 x - 1324838279 \)

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:  \(42867551195672722495351149174628212645190717310766431451792643783547492533=11^{18}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.00$
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:  $11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(407=11\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{407}(384,·)$, $\chi_{407}(1,·)$, $\chi_{407}(386,·)$, $\chi_{407}(131,·)$, $\chi_{407}(12,·)$, $\chi_{407}(397,·)$, $\chi_{407}(142,·)$, $\chi_{407}(144,·)$, $\chi_{407}(274,·)$, $\chi_{407}(153,·)$, $\chi_{407}(155,·)$, $\chi_{407}(287,·)$, $\chi_{407}(32,·)$, $\chi_{407}(34,·)$, $\chi_{407}(43,·)$, $\chi_{407}(54,·)$, $\chi_{407}(188,·)$, $\chi_{407}(318,·)$, $\chi_{407}(67,·)$, $\chi_{407}(76,·)$, $\chi_{407}(78,·)$, $\chi_{407}(208,·)$, $\chi_{407}(210,·)$, $\chi_{407}(342,·)$, $\chi_{407}(87,·)$, $\chi_{407}(221,·)$, $\chi_{407}(351,·)$, $\chi_{407}(98,·)$, $\chi_{407}(100,·)$, $\chi_{407}(230,·)$, $\chi_{407}(232,·)$, $\chi_{407}(362,·)$, $\chi_{407}(109,·)$, $\chi_{407}(241,·)$, $\chi_{407}(243,·)$, $\chi_{407}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{64624199} a^{19} - \frac{6183736}{64624199} a^{18} - \frac{57}{64624199} a^{17} + \frac{10800749}{64624199} a^{16} + \frac{1368}{64624199} a^{15} - \frac{16832156}{64624199} a^{14} - \frac{17955}{64624199} a^{13} - \frac{24108677}{64624199} a^{12} + \frac{140049}{64624199} a^{11} - \frac{9541767}{64624199} a^{10} - \frac{660231}{64624199} a^{9} + \frac{14779571}{64624199} a^{8} + \frac{1828332}{64624199} a^{7} - \frac{12944266}{64624199} a^{6} - \frac{2742498}{64624199} a^{5} + \frac{10933285}{64624199} a^{4} + \frac{1869885}{64624199} a^{3} - \frac{21076028}{64624199} a^{2} - \frac{373977}{64624199} a - \frac{10406257}{64624199}$, $\frac{1}{64624199} a^{20} - \frac{60}{64624199} a^{18} - \frac{18551208}{64624199} a^{17} + \frac{1530}{64624199} a^{16} - \frac{23251377}{64624199} a^{15} - \frac{21600}{64624199} a^{14} - \frac{28714675}{64624199} a^{13} + \frac{184275}{64624199} a^{12} - \frac{12389502}{64624199} a^{11} - \frac{972972}{64624199} a^{10} + \frac{18972579}{64624199} a^{9} + \frac{3127410}{64624199} a^{8} - \frac{29526765}{64624199} a^{7} - \frac{5773680}{64624199} a^{6} + \frac{3494934}{64624199} a^{5} + \frac{5412825}{64624199} a^{4} - \frac{30691743}{64624199} a^{3} - \frac{1968300}{64624199} a^{2} - \frac{8483114}{64624199} a + \frac{118098}{64624199}$, $\frac{1}{64624199} a^{21} - \frac{1830174}{64624199} a^{18} - \frac{1890}{64624199} a^{17} - \frac{21448427}{64624199} a^{16} + \frac{60480}{64624199} a^{15} - \frac{4656851}{64624199} a^{14} - \frac{893025}{64624199} a^{13} + \frac{27446455}{64624199} a^{12} + \frac{7429968}{64624199} a^{11} + \frac{28084350}{64624199} a^{10} + \frac{28137749}{64624199} a^{9} + \frac{17132908}{64624199} a^{8} - \frac{25322158}{64624199} a^{7} + \frac{2329362}{64624199} a^{6} - \frac{29888657}{64624199} a^{5} - \frac{20936633}{64624199} a^{4} - \frac{19023598}{64624199} a^{3} + \frac{19439186}{64624199} a^{2} - \frac{22320522}{64624199} a + \frac{21866570}{64624199}$, $\frac{1}{64624199} a^{22} - \frac{2079}{64624199} a^{18} + \frac{3480053}{64624199} a^{17} + \frac{70686}{64624199} a^{16} - \frac{21322580}{64624199} a^{15} - \frac{1122660}{64624199} a^{14} - \frac{4234623}{64624199} a^{13} + \frac{10216206}{64624199} a^{12} - \frac{22074557}{64624199} a^{11} + \frac{8435066}{64624199} a^{10} + \frac{22795616}{64624199} a^{9} - \frac{8104443}{64624199} a^{8} - \frac{8380891}{64624199} a^{7} - \frac{26980526}{64624199} a^{6} + \frac{375902}{885263} a^{5} + \frac{10309025}{64624199} a^{4} - \frac{4733068}{64624199} a^{3} + \frac{6485527}{64624199} a^{2} + \frac{13776181}{64624199} a + \frac{7440174}{64624199}$, $\frac{1}{64624199} a^{23} + \frac{7708510}{64624199} a^{18} - \frac{47817}{64624199} a^{17} + \frac{8837538}{64624199} a^{16} + \frac{1721412}{64624199} a^{15} + \frac{28028911}{64624199} a^{14} - \frac{27112239}{64624199} a^{13} + \frac{4364384}{64624199} a^{12} - \frac{23524058}{64624199} a^{11} + \frac{25091116}{64624199} a^{10} - \frac{23616513}{64624199} a^{9} + \frac{21852693}{64624199} a^{8} + \frac{25918160}{64624199} a^{7} - \frac{21384}{64624199} a^{6} - \frac{4414805}{64624199} a^{5} - \frac{22151601}{64624199} a^{4} + \frac{16524502}{64624199} a^{3} + \frac{11920891}{64624199} a^{2} + \frac{5432379}{64624199} a + \frac{14498362}{64624199}$, $\frac{1}{64624199} a^{24} - \frac{54648}{64624199} a^{18} - \frac{4146785}{64624199} a^{17} + \frac{2090286}{64624199} a^{16} + \frac{16531668}{64624199} a^{15} + \frac{29212295}{64624199} a^{14} - \frac{196888}{885263} a^{13} + \frac{12554345}{64624199} a^{12} + \frac{3218421}{64624199} a^{11} - \frac{24861581}{64624199} a^{10} + \frac{4950457}{64624199} a^{9} + \frac{11205612}{64624199} a^{8} - \frac{17839391}{64624199} a^{7} + \frac{8373074}{64624199} a^{6} - \frac{27736690}{64624199} a^{5} + \frac{5774007}{64624199} a^{4} + \frac{24541297}{64624199} a^{3} - \frac{5885948}{64624199} a^{2} - \frac{950559}{64624199} a + \frac{10412350}{64624199}$, $\frac{1}{64624199} a^{25} - \frac{13015142}{64624199} a^{18} - \frac{1024650}{64624199} a^{17} - \frac{21570646}{64624199} a^{16} - \frac{25277639}{64624199} a^{15} + \frac{2814654}{64624199} a^{14} + \frac{712490}{64624199} a^{13} + \frac{5782738}{64624199} a^{12} + \frac{2880689}{64624199} a^{11} + \frac{19129172}{64624199} a^{10} - \frac{8795034}{64624199} a^{9} - \frac{17082485}{64624199} a^{8} + \frac{14048556}{64624199} a^{7} - \frac{404148}{885263} a^{6} - \frac{2739216}{64624199} a^{5} - \frac{8643977}{64624199} a^{4} + \frac{8730913}{64624199} a^{3} - \frac{31254125}{64624199} a^{2} - \frac{5435862}{64624199} a + \frac{11818664}{64624199}$, $\frac{1}{64624199} a^{26} - \frac{1210950}{64624199} a^{18} + \frac{12056648}{64624199} a^{17} - \frac{15217439}{64624199} a^{16} - \frac{28750014}{64624199} a^{15} - \frac{31769413}{64624199} a^{14} + \frac{11712}{64624199} a^{13} - \frac{29525268}{64624199} a^{12} - \frac{13405864}{64624199} a^{11} + \frac{21872765}{64624199} a^{10} - \frac{2183456}{64624199} a^{9} - \frac{28832195}{64624199} a^{8} - \frac{16079639}{64624199} a^{7} + \frac{11150271}{64624199} a^{6} + \frac{3533375}{64624199} a^{5} - \frac{20092682}{64624199} a^{4} + \frac{25067334}{64624199} a^{3} - \frac{8621294}{64624199} a^{2} + \frac{13479212}{64624199} a + \frac{31351313}{64624199}$, $\frac{1}{64624199} a^{27} + \frac{16758175}{64624199} a^{18} - \frac{19617390}{64624199} a^{17} - \frac{24135676}{64624199} a^{16} + \frac{9205212}{64624199} a^{15} + \frac{25437505}{64624199} a^{14} + \frac{6222545}{64624199} a^{13} + \frac{18448629}{64624199} a^{12} - \frac{24313060}{64624199} a^{11} + \frac{7976497}{64624199} a^{10} - \frac{4971617}{64624199} a^{9} + \frac{21254955}{64624199} a^{8} + \frac{4727931}{64624199} a^{7} + \frac{2584921}{64624199} a^{6} - \frac{10459172}{64624199} a^{5} - \frac{2359444}{64624199} a^{4} + \frac{25934894}{64624199} a^{3} + \frac{32283682}{64624199} a^{2} - \frac{14334444}{64624199} a + \frac{3394054}{64624199}$, $\frac{1}{64624199} a^{28} - \frac{23882040}{64624199} a^{18} + \frac{26341513}{64624199} a^{17} - \frac{19000484}{64624199} a^{16} - \frac{22779449}{64624199} a^{15} - \frac{5391285}{64624199} a^{14} + \frac{21210210}{64624199} a^{13} + \frac{30435812}{64624199} a^{12} - \frac{638995}{64624199} a^{11} - \frac{32075246}{64624199} a^{10} - \frac{21217410}{64624199} a^{9} + \frac{7948207}{64624199} a^{8} - \frac{9047697}{64624199} a^{7} + \frac{25109650}{64624199} a^{6} + \frac{15089483}{64624199} a^{5} + \frac{5342829}{64624199} a^{4} - \frac{6050486}{64624199} a^{3} + \frac{11088035}{64624199} a^{2} - \frac{14788792}{64624199} a - \frac{11457102}{64624199}$, $\frac{1}{64624199} a^{29} - \frac{15242142}{64624199} a^{18} - \frac{23168585}{64624199} a^{17} + \frac{25515155}{64624199} a^{16} + \frac{30018940}{64624199} a^{15} - \frac{31431196}{64624199} a^{14} + \frac{9967977}{64624199} a^{13} + \frac{451301}{64624199} a^{12} - \frac{1674531}{64624199} a^{11} - \frac{1231275}{64624199} a^{10} + \frac{3110977}{64624199} a^{9} - \frac{27442828}{64624199} a^{8} + \frac{13649595}{64624199} a^{7} + \frac{14928656}{64624199} a^{6} - \frac{9779188}{64624199} a^{5} - \frac{20969462}{64624199} a^{4} + \frac{836256}{64624199} a^{3} + \frac{4849587}{64624199} a^{2} - \frac{22331586}{64624199} a - \frac{13422139}{64624199}$, $\frac{1}{64624199} a^{30} + \frac{19507418}{64624199} a^{18} - \frac{3172352}{64624199} a^{17} - \frac{12638257}{64624199} a^{16} + \frac{10826982}{64624199} a^{15} - \frac{31920175}{64624199} a^{14} + \frac{11274456}{64624199} a^{13} - \frac{14733492}{64624199} a^{12} - \frac{21027685}{64624199} a^{11} - \frac{10711840}{64624199} a^{10} - \frac{235687}{885263} a^{9} + \frac{1472159}{64624199} a^{8} + \frac{11433627}{64624199} a^{7} - \frac{24447970}{64624199} a^{6} + \frac{16585181}{64624199} a^{5} + \frac{2874630}{64624199} a^{4} - \frac{23693315}{64624199} a^{3} + \frac{21988085}{64624199} a^{2} + \frac{18136121}{64624199} a - \frac{12856894}{64624199}$, $\frac{1}{64624199} a^{31} + \frac{26692314}{64624199} a^{18} + \frac{673186}{64624199} a^{17} + \frac{13739993}{64624199} a^{16} - \frac{28273812}{64624199} a^{15} - \frac{21956192}{64624199} a^{14} - \frac{22201882}{64624199} a^{13} - \frac{1266070}{64624199} a^{12} - \frac{17082597}{64624199} a^{11} + \frac{13844526}{64624199} a^{10} - \frac{5422386}{64624199} a^{9} - \frac{23091202}{64624199} a^{8} - \frac{30210845}{64624199} a^{7} - \frac{10563883}{64624199} a^{6} - \frac{20793157}{64624199} a^{5} + \frac{29021240}{64624199} a^{4} + \frac{5813113}{64624199} a^{3} + \frac{10570621}{64624199} a^{2} + \frac{16227780}{64624199} a - \frac{9013548}{64624199}$, $\frac{1}{64624199} a^{32} + \frac{6031614}{64624199} a^{18} - \frac{15778885}{64624199} a^{17} + \frac{5492071}{64624199} a^{16} - \frac{24369309}{64624199} a^{15} + \frac{25868238}{64624199} a^{14} + \frac{6172016}{64624199} a^{13} + \frac{24367204}{64624199} a^{12} - \frac{31247705}{64624199} a^{11} - \frac{16954826}{64624199} a^{10} - \frac{13618167}{64624199} a^{9} + \frac{27635321}{64624199} a^{8} + \frac{27827296}{64624199} a^{7} + \frac{16563653}{64624199} a^{6} + \frac{3746571}{64624199} a^{5} + \frac{4952350}{64624199} a^{4} - \frac{16258604}{64624199} a^{3} - \frac{25390606}{64624199} a^{2} - \frac{3647703}{64624199} a - \frac{6491112}{64624199}$, $\frac{1}{64624199} a^{33} - \frac{28226030}{64624199} a^{18} + \frac{26173074}{64624199} a^{17} + \frac{1534531}{64624199} a^{16} - \frac{18106441}{64624199} a^{15} + \frac{4953407}{64624199} a^{14} + \frac{11839050}{64624199} a^{13} - \frac{3537076}{64624199} a^{12} + \frac{29065416}{64624199} a^{11} - \frac{1851261}{64624199} a^{10} + \frac{13787377}{64624199} a^{9} - \frac{16079529}{64624199} a^{8} + \frac{20114160}{64624199} a^{7} - \frac{1511169}{64624199} a^{6} + \frac{31938689}{64624199} a^{5} + \frac{4993762}{64624199} a^{4} + \frac{23771280}{64624199} a^{3} + \frac{6175992}{64624199} a^{2} - \frac{29248329}{64624199} a + \frac{13633252}{64624199}$, $\frac{1}{64624199} a^{34} - \frac{20455687}{64624199} a^{18} + \frac{8255796}{64624199} a^{17} - \frac{20497108}{64624199} a^{16} - \frac{27108555}{64624199} a^{15} - \frac{31307241}{64624199} a^{14} - \frac{18937168}{64624199} a^{13} + \frac{22655315}{64624199} a^{12} + \frac{27795578}{64624199} a^{11} + \frac{8339792}{64624199} a^{10} + \frac{28797370}{64624199} a^{9} - \frac{21881609}{64624199} a^{8} - \frac{6479445}{64624199} a^{7} - \frac{20860981}{64624199} a^{6} + \frac{12446176}{64624199} a^{5} + \frac{20860981}{64624199} a^{4} + \frac{12844655}{64624199} a^{3} + \frac{7604615}{64624199} a^{2} - \frac{26475000}{64624199} a + \frac{2057130}{64624199}$, $\frac{1}{64624199} a^{35} + \frac{3807008}{64624199} a^{18} - \frac{23235685}{64624199} a^{17} + \frac{19253400}{64624199} a^{16} - \frac{30205592}{64624199} a^{15} + \frac{4218128}{64624199} a^{14} + \frac{118147}{64624199} a^{13} + \frac{18266289}{64624199} a^{12} + \frac{16106785}{64624199} a^{11} - \frac{753242}{64624199} a^{10} - \frac{12337291}{64624199} a^{9} - \frac{12979147}{64624199} a^{8} - \frac{2551570}{64624199} a^{7} - \frac{15470652}{64624199} a^{6} + \frac{25708964}{64624199} a^{5} - \frac{4881805}{64624199} a^{4} + \frac{18986490}{64624199} a^{3} - \frac{18498093}{64624199} a^{2} - \frac{219245}{64624199} a - \frac{590882}{64624199}$

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:  $35$
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:  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

$C_{36}$ (as 36T1):

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

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.4.6129013.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.315191919957242668714693.1, \(\Q(\zeta_{37})^+\)

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 $36$ $18^{2}$ $36$ $18^{2}$ R $36$ $36$ $36$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$ $36$

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
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37Data not computed