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Properties

Label 36.0.44387950739...5625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 5^{18}\cdot 19^{34}$
Root discriminant $62.48$
Ramified primes $3, 5, 19$
Class number $247608$ (GRH)
Class group $[247608]$ (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21850951, -5418330, -4108065, 3047715, 267795, 2102034, -560238, 5476374, -1026513, 5237843, -276502, 2800770, 1345047, 517817, 3063059, -539957, 3956632, -809761, 3665766, -711664, 2542252, -457647, 1332275, -218657, 527433, -77194, 156466, -19818, 34162, -3586, 5314, -432, 556, -31, 35, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 35*x^34 - 31*x^33 + 556*x^32 - 432*x^31 + 5314*x^30 - 3586*x^29 + 34162*x^28 - 19818*x^27 + 156466*x^26 - 77194*x^25 + 527433*x^24 - 218657*x^23 + 1332275*x^22 - 457647*x^21 + 2542252*x^20 - 711664*x^19 + 3665766*x^18 - 809761*x^17 + 3956632*x^16 - 539957*x^15 + 3063059*x^14 + 517817*x^13 + 1345047*x^12 + 2800770*x^11 - 276502*x^10 + 5237843*x^9 - 1026513*x^8 + 5476374*x^7 - 560238*x^6 + 2102034*x^5 + 267795*x^4 + 3047715*x^3 - 4108065*x^2 - 5418330*x + 21850951)
 
gp: K = bnfinit(x^36 - x^35 + 35*x^34 - 31*x^33 + 556*x^32 - 432*x^31 + 5314*x^30 - 3586*x^29 + 34162*x^28 - 19818*x^27 + 156466*x^26 - 77194*x^25 + 527433*x^24 - 218657*x^23 + 1332275*x^22 - 457647*x^21 + 2542252*x^20 - 711664*x^19 + 3665766*x^18 - 809761*x^17 + 3956632*x^16 - 539957*x^15 + 3063059*x^14 + 517817*x^13 + 1345047*x^12 + 2800770*x^11 - 276502*x^10 + 5237843*x^9 - 1026513*x^8 + 5476374*x^7 - 560238*x^6 + 2102034*x^5 + 267795*x^4 + 3047715*x^3 - 4108065*x^2 - 5418330*x + 21850951, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 35 x^{34} - 31 x^{33} + 556 x^{32} - 432 x^{31} + 5314 x^{30} - 3586 x^{29} + 34162 x^{28} - 19818 x^{27} + 156466 x^{26} - 77194 x^{25} + 527433 x^{24} - 218657 x^{23} + 1332275 x^{22} - 457647 x^{21} + 2542252 x^{20} - 711664 x^{19} + 3665766 x^{18} - 809761 x^{17} + 3956632 x^{16} - 539957 x^{15} + 3063059 x^{14} + 517817 x^{13} + 1345047 x^{12} + 2800770 x^{11} - 276502 x^{10} + 5237843 x^{9} - 1026513 x^{8} + 5476374 x^{7} - 560238 x^{6} + 2102034 x^{5} + 267795 x^{4} + 3047715 x^{3} - 4108065 x^{2} - 5418330 x + 21850951 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44387950739803436916061690678602018428037077267652774810791015625=3^{18}\cdot 5^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.48$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(259,·)$, $\chi_{285}(134,·)$, $\chi_{285}(109,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(146,·)$, $\chi_{285}(149,·)$, $\chi_{285}(281,·)$, $\chi_{285}(154,·)$, $\chi_{285}(34,·)$, $\chi_{285}(41,·)$, $\chi_{285}(44,·)$, $\chi_{285}(184,·)$, $\chi_{285}(61,·)$, $\chi_{285}(194,·)$, $\chi_{285}(196,·)$, $\chi_{285}(71,·)$, $\chi_{285}(74,·)$, $\chi_{285}(79,·)$, $\chi_{285}(56,·)$, $\chi_{285}(86,·)$, $\chi_{285}(221,·)$, $\chi_{285}(94,·)$, $\chi_{285}(226,·)$, $\chi_{285}(104,·)$, $\chi_{285}(106,·)$, $\chi_{285}(236,·)$, $\chi_{285}(274,·)$, $\chi_{285}(239,·)$, $\chi_{285}(116,·)$, $\chi_{285}(119,·)$, $\chi_{285}(121,·)$, $\chi_{285}(124,·)$, $\chi_{285}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4181} a^{19} + \frac{19}{4181} a^{17} + \frac{152}{4181} a^{15} + \frac{665}{4181} a^{13} + \frac{1729}{4181} a^{11} - \frac{1464}{4181} a^{9} - \frac{1673}{4181} a^{7} + \frac{1254}{4181} a^{5} + \frac{285}{4181} a^{3} + \frac{19}{4181} a - \frac{1597}{4181}$, $\frac{1}{4181} a^{20} + \frac{19}{4181} a^{18} + \frac{152}{4181} a^{16} + \frac{665}{4181} a^{14} + \frac{1729}{4181} a^{12} - \frac{1464}{4181} a^{10} - \frac{1673}{4181} a^{8} + \frac{1254}{4181} a^{6} + \frac{285}{4181} a^{4} + \frac{19}{4181} a^{2} - \frac{1597}{4181} a$, $\frac{1}{4181} a^{21} - \frac{209}{4181} a^{17} + \frac{1958}{4181} a^{15} + \frac{1637}{4181} a^{13} - \frac{867}{4181} a^{11} + \frac{1057}{4181} a^{9} - \frac{11}{113} a^{7} + \frac{1545}{4181} a^{5} - \frac{1215}{4181} a^{3} - \frac{1597}{4181} a^{2} - \frac{361}{4181} a + \frac{1076}{4181}$, $\frac{1}{4181} a^{22} - \frac{209}{4181} a^{18} + \frac{1958}{4181} a^{16} + \frac{1637}{4181} a^{14} - \frac{867}{4181} a^{12} + \frac{1057}{4181} a^{10} - \frac{11}{113} a^{8} + \frac{1545}{4181} a^{6} - \frac{1215}{4181} a^{4} - \frac{1597}{4181} a^{3} - \frac{361}{4181} a^{2} + \frac{1076}{4181} a$, $\frac{1}{4181} a^{23} + \frac{1748}{4181} a^{17} - \frac{43}{4181} a^{15} + \frac{145}{4181} a^{13} - \frac{1329}{4181} a^{11} - \frac{1170}{4181} a^{9} - \frac{1089}{4181} a^{7} + \frac{1649}{4181} a^{5} - \frac{1597}{4181} a^{4} + \frac{670}{4181} a^{3} + \frac{1076}{4181} a^{2} - \frac{210}{4181} a + \frac{707}{4181}$, $\frac{1}{4181} a^{24} + \frac{1748}{4181} a^{18} - \frac{43}{4181} a^{16} + \frac{145}{4181} a^{14} - \frac{1329}{4181} a^{12} - \frac{1170}{4181} a^{10} - \frac{1089}{4181} a^{8} + \frac{1649}{4181} a^{6} - \frac{1597}{4181} a^{5} + \frac{670}{4181} a^{4} + \frac{1076}{4181} a^{3} - \frac{210}{4181} a^{2} + \frac{707}{4181} a$, $\frac{1}{4181} a^{25} + \frac{193}{4181} a^{17} + \frac{2033}{4181} a^{15} - \frac{1431}{4181} a^{13} - \frac{599}{4181} a^{11} - \frac{789}{4181} a^{9} - \frac{647}{4181} a^{7} - \frac{1597}{4181} a^{6} - \frac{478}{4181} a^{5} + \frac{1076}{4181} a^{4} - \frac{23}{113} a^{3} + \frac{707}{4181} a^{2} + \frac{236}{4181} a - \frac{1352}{4181}$, $\frac{1}{4181} a^{26} + \frac{193}{4181} a^{18} + \frac{2033}{4181} a^{16} - \frac{1431}{4181} a^{14} - \frac{599}{4181} a^{12} - \frac{789}{4181} a^{10} - \frac{647}{4181} a^{8} - \frac{1597}{4181} a^{7} - \frac{478}{4181} a^{6} + \frac{1076}{4181} a^{5} - \frac{23}{113} a^{4} + \frac{707}{4181} a^{3} + \frac{236}{4181} a^{2} - \frac{1352}{4181} a$, $\frac{1}{8362} a^{27} - \frac{1}{8362} a^{25} - \frac{1}{8362} a^{24} + \frac{2433}{8362} a^{18} + \frac{1177}{4181} a^{17} - \frac{2069}{4181} a^{16} - \frac{3533}{8362} a^{15} - \frac{145}{8362} a^{14} - \frac{2083}{8362} a^{13} + \frac{1329}{8362} a^{12} + \frac{593}{8362} a^{11} - \frac{3011}{8362} a^{10} + \frac{2567}{8362} a^{9} + \frac{3673}{8362} a^{8} + \frac{1121}{8362} a^{7} - \frac{3157}{8362} a^{6} - \frac{2481}{8362} a^{5} - \frac{1039}{8362} a^{4} - \frac{641}{8362} a^{3} + \frac{1166}{4181} a^{2} - \frac{429}{8362} a + \frac{179}{8362}$, $\frac{1}{365435296162} a^{28} + \frac{10307271}{365435296162} a^{27} + \frac{33400433}{365435296162} a^{26} - \frac{18960343}{182717648081} a^{25} - \frac{8115083}{365435296162} a^{24} + \frac{12658451}{182717648081} a^{23} + \frac{4584315}{182717648081} a^{22} - \frac{10507422}{182717648081} a^{21} - \frac{380437}{182717648081} a^{20} - \frac{28367813}{365435296162} a^{19} + \frac{130291055613}{365435296162} a^{18} + \frac{10984283954}{182717648081} a^{17} - \frac{133758775925}{365435296162} a^{16} + \frac{29380810762}{182717648081} a^{15} - \frac{43422360839}{182717648081} a^{14} + \frac{17821937280}{182717648081} a^{13} - \frac{90949798717}{182717648081} a^{12} - \frac{53150887485}{182717648081} a^{11} - \frac{72220098319}{182717648081} a^{10} - \frac{51564015211}{182717648081} a^{9} + \frac{74081372186}{182717648081} a^{8} - \frac{15787791250}{182717648081} a^{7} - \frac{49857197532}{182717648081} a^{6} + \frac{53995906080}{182717648081} a^{5} + \frac{41175341015}{182717648081} a^{4} - \frac{166833076499}{365435296162} a^{3} + \frac{138685955917}{365435296162} a^{2} + \frac{15868408690}{182717648081} a - \frac{145352519837}{365435296162}$, $\frac{1}{365435296162} a^{29} - \frac{15097}{365435296162} a^{27} - \frac{33398201}{365435296162} a^{26} + \frac{16514888}{182717648081} a^{25} + \frac{19509460}{182717648081} a^{24} - \frac{888059}{182717648081} a^{23} + \frac{18856443}{182717648081} a^{22} + \frac{7992507}{182717648081} a^{21} + \frac{25204987}{365435296162} a^{20} - \frac{13748555}{182717648081} a^{19} + \frac{53916655076}{182717648081} a^{18} - \frac{35770113307}{365435296162} a^{17} + \frac{110511607349}{365435296162} a^{16} + \frac{80941420161}{365435296162} a^{15} - \frac{89676440103}{365435296162} a^{14} - \frac{25581325409}{365435296162} a^{13} - \frac{75556967433}{365435296162} a^{12} + \frac{64132960559}{365435296162} a^{11} + \frac{16725324389}{365435296162} a^{10} + \frac{18265361069}{365435296162} a^{9} - \frac{73958534237}{365435296162} a^{8} - \frac{108409572551}{365435296162} a^{7} + \frac{168735927177}{365435296162} a^{6} - \frac{87646735033}{365435296162} a^{5} - \frac{63292603047}{182717648081} a^{4} + \frac{54746252253}{365435296162} a^{3} - \frac{127179139099}{365435296162} a^{2} + \frac{74666763508}{182717648081} a + \frac{88487993856}{182717648081}$, $\frac{1}{365435296162} a^{30} - \frac{1647737}{182717648081} a^{27} - \frac{40570763}{365435296162} a^{26} - \frac{568626}{4938314813} a^{25} + \frac{24946235}{365435296162} a^{24} + \frac{15078117}{182717648081} a^{23} - \frac{6415122}{182717648081} a^{22} + \frac{39906379}{365435296162} a^{21} + \frac{309324}{4938314813} a^{20} - \frac{357421}{9876629626} a^{19} + \frac{21287939385}{182717648081} a^{18} - \frac{84424503967}{365435296162} a^{17} + \frac{47531490237}{182717648081} a^{16} - \frac{150863592917}{365435296162} a^{15} + \frac{165333847857}{365435296162} a^{14} - \frac{154487393441}{365435296162} a^{13} - \frac{171947611583}{365435296162} a^{12} + \frac{30104032187}{365435296162} a^{11} - \frac{95063750155}{365435296162} a^{10} + \frac{98364104571}{365435296162} a^{9} - \frac{125342615079}{365435296162} a^{8} + \frac{154410445663}{365435296162} a^{7} - \frac{142475139751}{365435296162} a^{6} - \frac{50223399394}{182717648081} a^{5} + \frac{150850670983}{365435296162} a^{4} - \frac{18072717126}{182717648081} a^{3} + \frac{66269187435}{365435296162} a^{2} + \frac{29076288220}{182717648081} a + \frac{87185205827}{365435296162}$, $\frac{1}{365435296162} a^{31} + \frac{468441}{365435296162} a^{27} - \frac{6755371}{182717648081} a^{26} + \frac{24926195}{365435296162} a^{25} + \frac{6911817}{182717648081} a^{24} - \frac{16161997}{182717648081} a^{23} - \frac{35645391}{365435296162} a^{22} + \frac{14482904}{182717648081} a^{21} - \frac{13437077}{365435296162} a^{20} - \frac{10704003}{182717648081} a^{19} - \frac{167183694949}{365435296162} a^{18} - \frac{31998829735}{182717648081} a^{17} + \frac{22246833781}{365435296162} a^{16} + \frac{155408943639}{365435296162} a^{15} - \frac{102292189961}{365435296162} a^{14} - \frac{77869694539}{365435296162} a^{13} + \frac{31014448503}{365435296162} a^{12} - \frac{80882893211}{365435296162} a^{11} - \frac{129692031043}{365435296162} a^{10} + \frac{176968595605}{365435296162} a^{9} + \frac{88742309503}{365435296162} a^{8} - \frac{145943704947}{365435296162} a^{7} + \frac{51003789003}{182717648081} a^{6} + \frac{164158551963}{365435296162} a^{5} - \frac{29873012990}{182717648081} a^{4} - \frac{24574663531}{365435296162} a^{3} - \frac{2185891922}{4938314813} a^{2} - \frac{127868726861}{365435296162} a + \frac{11927423957}{182717648081}$, $\frac{1}{365435296162} a^{32} - \frac{27437}{9876629626} a^{27} - \frac{20058270}{182717648081} a^{26} + \frac{9395444}{182717648081} a^{25} + \frac{39115025}{365435296162} a^{24} - \frac{1033973}{9876629626} a^{23} + \frac{1093228}{182717648081} a^{22} - \frac{1714331}{365435296162} a^{21} - \frac{14767564}{182717648081} a^{20} - \frac{16390333}{182717648081} a^{19} - \frac{135259479445}{365435296162} a^{18} - \frac{51778336899}{365435296162} a^{17} - \frac{22060349955}{182717648081} a^{16} - \frac{166169740651}{365435296162} a^{15} + \frac{16783991211}{365435296162} a^{14} + \frac{9382028003}{365435296162} a^{13} - \frac{4576523801}{365435296162} a^{12} - \frac{2404352969}{9876629626} a^{11} - \frac{91626960519}{365435296162} a^{10} + \frac{106430131489}{365435296162} a^{9} - \frac{113892967313}{365435296162} a^{8} - \frac{9997009894}{182717648081} a^{7} - \frac{82141368387}{365435296162} a^{6} - \frac{56556430964}{182717648081} a^{5} + \frac{116421954435}{365435296162} a^{4} + \frac{93146005483}{365435296162} a^{3} + \frac{20124620988}{182717648081} a^{2} + \frac{11311359643}{182717648081} a + \frac{23644161323}{365435296162}$, $\frac{1}{365435296162} a^{33} - \frac{3990536}{182717648081} a^{27} + \frac{21626393}{365435296162} a^{26} - \frac{20565668}{182717648081} a^{25} - \frac{17295419}{365435296162} a^{24} + \frac{102223}{1616970337} a^{23} - \frac{11044643}{365435296162} a^{22} - \frac{20153901}{182717648081} a^{21} + \frac{13161852}{182717648081} a^{20} - \frac{7927490}{182717648081} a^{19} + \frac{70318349763}{365435296162} a^{18} + \frac{52494685136}{182717648081} a^{17} + \frac{58467733199}{182717648081} a^{16} + \frac{35573049809}{182717648081} a^{15} - \frac{76240293942}{182717648081} a^{14} + \frac{54226677928}{182717648081} a^{13} + \frac{80874868200}{182717648081} a^{12} - \frac{76506392377}{182717648081} a^{11} + \frac{13575260029}{182717648081} a^{10} + \frac{74233031043}{182717648081} a^{9} - \frac{56108378907}{365435296162} a^{8} + \frac{58107835816}{182717648081} a^{7} + \frac{98724277311}{365435296162} a^{6} - \frac{55612683393}{182717648081} a^{5} - \frac{80229680599}{182717648081} a^{4} + \frac{13946186581}{182717648081} a^{3} - \frac{163211952731}{365435296162} a^{2} - \frac{58868345731}{182717648081} a + \frac{77136253202}{182717648081}$, $\frac{1}{365435296162} a^{34} - \frac{5041282}{182717648081} a^{27} + \frac{12212639}{182717648081} a^{26} + \frac{13650383}{182717648081} a^{25} + \frac{9011541}{365435296162} a^{24} + \frac{5149395}{365435296162} a^{23} + \frac{15693668}{182717648081} a^{22} + \frac{17214289}{182717648081} a^{21} + \frac{17661724}{182717648081} a^{20} - \frac{35273703}{365435296162} a^{19} - \frac{115962727739}{365435296162} a^{18} + \frac{43528677389}{182717648081} a^{17} - \frac{42466641681}{182717648081} a^{16} + \frac{70861615145}{365435296162} a^{15} - \frac{118947655865}{365435296162} a^{14} - \frac{140189473137}{365435296162} a^{13} - \frac{73946875991}{365435296162} a^{12} + \frac{23809824887}{365435296162} a^{11} - \frac{1460546197}{3233940674} a^{10} - \frac{41403744175}{182717648081} a^{9} + \frac{163761817011}{365435296162} a^{8} + \frac{73274338184}{182717648081} a^{7} + \frac{7269138549}{365435296162} a^{6} - \frac{76067201611}{365435296162} a^{5} - \frac{103003026713}{365435296162} a^{4} + \frac{17373816518}{182717648081} a^{3} + \frac{14280520541}{182717648081} a^{2} - \frac{137694741245}{365435296162} a + \frac{19570887}{3233940674}$, $\frac{1}{365435296162} a^{35} + \frac{5771754}{182717648081} a^{27} - \frac{5622956}{182717648081} a^{26} - \frac{28903197}{365435296162} a^{25} - \frac{41819969}{365435296162} a^{24} + \frac{4247572}{182717648081} a^{23} + \frac{1390091}{182717648081} a^{22} + \frac{17958807}{182717648081} a^{21} + \frac{10434505}{365435296162} a^{20} + \frac{21073585}{365435296162} a^{19} + \frac{36464091160}{182717648081} a^{18} + \frac{72012319006}{182717648081} a^{17} + \frac{1771711269}{9876629626} a^{16} + \frac{80063210587}{365435296162} a^{15} - \frac{22448524915}{365435296162} a^{14} - \frac{1808732873}{365435296162} a^{13} - \frac{165197095637}{365435296162} a^{12} + \frac{2399779383}{9876629626} a^{11} + \frac{59949620957}{182717648081} a^{10} - \frac{32700102941}{365435296162} a^{9} - \frac{54142927042}{182717648081} a^{8} + \frac{145131403053}{365435296162} a^{7} - \frac{98891290379}{365435296162} a^{6} + \frac{9219638105}{365435296162} a^{5} + \frac{26718983736}{182717648081} a^{4} + \frac{18673658224}{182717648081} a^{3} - \frac{132545056761}{365435296162} a^{2} + \frac{105945550113}{365435296162} a - \frac{17207018329}{182717648081}$

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

Class group and class number

$C_{247608}$, which has order $247608$ (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:  $17$
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:  \( 1438232971979.9597 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{57}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-95}) \), 3.3.361.1, \(\Q(\sqrt{-15}, \sqrt{57})\), 6.6.66854673.1, 6.0.439833375.1, 6.0.309512375.1, \(\Q(\zeta_{19})^+\), 12.0.69836676592764515625.3, \(\Q(\zeta_{57})^+\), 18.0.11088656920413061413017818359375.2, 18.0.10703880581610941769412109375.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{4}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$ $18^{2}$

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
3Data not computed
5Data not computed
19Data not computed