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Properties

Label 36.36.1040838713...1533.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{90}\cdot 13^{27}$
Root discriminant $106.72$
Ramified primes $3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-3561, -30861, 2832975, -24720957, 30525471, 221243481, -465058575, -821976822, 2198845116, 1686910755, -5555566935, -2137188807, 8756851740, 1763581788, -9272415006, -971296587, 6873288795, 358455213, -3650721350, -87330726, 1405474902, 13538481, -393214770, -1248885, 79561287, 60876, -11505240, -1187, 1165662, 0, -80130, 0, 3537, 0, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 90*x^34 + 3537*x^32 - 80130*x^30 + 1165662*x^28 - 1187*x^27 - 11505240*x^26 + 60876*x^25 + 79561287*x^24 - 1248885*x^23 - 393214770*x^22 + 13538481*x^21 + 1405474902*x^20 - 87330726*x^19 - 3650721350*x^18 + 358455213*x^17 + 6873288795*x^16 - 971296587*x^15 - 9272415006*x^14 + 1763581788*x^13 + 8756851740*x^12 - 2137188807*x^11 - 5555566935*x^10 + 1686910755*x^9 + 2198845116*x^8 - 821976822*x^7 - 465058575*x^6 + 221243481*x^5 + 30525471*x^4 - 24720957*x^3 + 2832975*x^2 - 30861*x - 3561)
 
gp: K = bnfinit(x^36 - 90*x^34 + 3537*x^32 - 80130*x^30 + 1165662*x^28 - 1187*x^27 - 11505240*x^26 + 60876*x^25 + 79561287*x^24 - 1248885*x^23 - 393214770*x^22 + 13538481*x^21 + 1405474902*x^20 - 87330726*x^19 - 3650721350*x^18 + 358455213*x^17 + 6873288795*x^16 - 971296587*x^15 - 9272415006*x^14 + 1763581788*x^13 + 8756851740*x^12 - 2137188807*x^11 - 5555566935*x^10 + 1686910755*x^9 + 2198845116*x^8 - 821976822*x^7 - 465058575*x^6 + 221243481*x^5 + 30525471*x^4 - 24720957*x^3 + 2832975*x^2 - 30861*x - 3561, 1)
 

Normalized defining polynomial

\( x^{36} - 90 x^{34} + 3537 x^{32} - 80130 x^{30} + 1165662 x^{28} - 1187 x^{27} - 11505240 x^{26} + 60876 x^{25} + 79561287 x^{24} - 1248885 x^{23} - 393214770 x^{22} + 13538481 x^{21} + 1405474902 x^{20} - 87330726 x^{19} - 3650721350 x^{18} + 358455213 x^{17} + 6873288795 x^{16} - 971296587 x^{15} - 9272415006 x^{14} + 1763581788 x^{13} + 8756851740 x^{12} - 2137188807 x^{11} - 5555566935 x^{10} + 1686910755 x^{9} + 2198845116 x^{8} - 821976822 x^{7} - 465058575 x^{6} + 221243481 x^{5} + 30525471 x^{4} - 24720957 x^{3} + 2832975 x^{2} - 30861 x - 3561 \)

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:  \(10408387130780717493462032934955905615684680986496729607398257892671441533=3^{90}\cdot 13^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.72$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(351=3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{351}(1,·)$, $\chi_{351}(259,·)$, $\chi_{351}(5,·)$, $\chi_{351}(8,·)$, $\chi_{351}(142,·)$, $\chi_{351}(274,·)$, $\chi_{351}(278,·)$, $\chi_{351}(25,·)$, $\chi_{351}(281,·)$, $\chi_{351}(157,·)$, $\chi_{351}(161,·)$, $\chi_{351}(164,·)$, $\chi_{351}(40,·)$, $\chi_{351}(64,·)$, $\chi_{351}(298,·)$, $\chi_{351}(44,·)$, $\chi_{351}(47,·)$, $\chi_{351}(181,·)$, $\chi_{351}(313,·)$, $\chi_{351}(317,·)$, $\chi_{351}(320,·)$, $\chi_{351}(196,·)$, $\chi_{351}(200,·)$, $\chi_{351}(203,·)$, $\chi_{351}(79,·)$, $\chi_{351}(337,·)$, $\chi_{351}(83,·)$, $\chi_{351}(86,·)$, $\chi_{351}(220,·)$, $\chi_{351}(103,·)$, $\chi_{351}(235,·)$, $\chi_{351}(239,·)$, $\chi_{351}(242,·)$, $\chi_{351}(118,·)$, $\chi_{351}(122,·)$, $\chi_{351}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{17} a^{27} - \frac{6}{17} a^{25} + \frac{1}{17} a^{23} + \frac{7}{17} a^{21} - \frac{3}{17} a^{19} - \frac{7}{17} a^{18} - \frac{7}{17} a^{17} - \frac{2}{17} a^{16} - \frac{2}{17} a^{15} + \frac{3}{17} a^{14} + \frac{1}{17} a^{13} + \frac{8}{17} a^{12} + \frac{2}{17} a^{11} + \frac{4}{17} a^{10} + \frac{8}{17} a^{9} + \frac{7}{17} a^{8} + \frac{7}{17} a^{7} - \frac{6}{17} a^{6} + \frac{8}{17} a^{5} + \frac{4}{17} a^{4} + \frac{6}{17} a^{3} + \frac{6}{17} a^{2} - \frac{2}{17} a + \frac{6}{17}$, $\frac{1}{17} a^{28} - \frac{6}{17} a^{26} + \frac{1}{17} a^{24} + \frac{7}{17} a^{22} - \frac{3}{17} a^{20} - \frac{7}{17} a^{19} - \frac{7}{17} a^{18} - \frac{2}{17} a^{17} - \frac{2}{17} a^{16} + \frac{3}{17} a^{15} + \frac{1}{17} a^{14} + \frac{8}{17} a^{13} + \frac{2}{17} a^{12} + \frac{4}{17} a^{11} + \frac{8}{17} a^{10} + \frac{7}{17} a^{9} + \frac{7}{17} a^{8} - \frac{6}{17} a^{7} + \frac{8}{17} a^{6} + \frac{4}{17} a^{5} + \frac{6}{17} a^{4} + \frac{6}{17} a^{3} - \frac{2}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{17} a^{29} - \frac{1}{17} a^{25} - \frac{4}{17} a^{23} + \frac{5}{17} a^{21} - \frac{7}{17} a^{20} - \frac{8}{17} a^{19} + \frac{7}{17} a^{18} + \frac{7}{17} a^{17} + \frac{8}{17} a^{16} + \frac{6}{17} a^{15} - \frac{8}{17} a^{14} + \frac{8}{17} a^{13} + \frac{1}{17} a^{12} + \frac{3}{17} a^{11} - \frac{3}{17} a^{10} + \frac{4}{17} a^{9} + \frac{2}{17} a^{8} - \frac{1}{17} a^{7} + \frac{2}{17} a^{6} + \frac{3}{17} a^{5} - \frac{4}{17} a^{4} + \frac{8}{17} a^{2} + \frac{5}{17} a + \frac{2}{17}$, $\frac{1}{17} a^{30} - \frac{1}{17} a^{26} - \frac{4}{17} a^{24} + \frac{5}{17} a^{22} - \frac{7}{17} a^{21} - \frac{8}{17} a^{20} + \frac{7}{17} a^{19} + \frac{7}{17} a^{18} + \frac{8}{17} a^{17} + \frac{6}{17} a^{16} - \frac{8}{17} a^{15} + \frac{8}{17} a^{14} + \frac{1}{17} a^{13} + \frac{3}{17} a^{12} - \frac{3}{17} a^{11} + \frac{4}{17} a^{10} + \frac{2}{17} a^{9} - \frac{1}{17} a^{8} + \frac{2}{17} a^{7} + \frac{3}{17} a^{6} - \frac{4}{17} a^{5} + \frac{8}{17} a^{3} + \frac{5}{17} a^{2} + \frac{2}{17} a$, $\frac{1}{1698317} a^{31} - \frac{34192}{1698317} a^{30} - \frac{29693}{1698317} a^{29} - \frac{43544}{1698317} a^{28} - \frac{29541}{1698317} a^{27} + \frac{539491}{1698317} a^{26} + \frac{8199}{1698317} a^{25} + \frac{117313}{1698317} a^{24} + \frac{438094}{1698317} a^{23} + \frac{837458}{1698317} a^{22} - \frac{515647}{1698317} a^{21} + \frac{314169}{1698317} a^{20} + \frac{310562}{1698317} a^{19} + \frac{40108}{1698317} a^{18} - \frac{692005}{1698317} a^{17} - \frac{517808}{1698317} a^{16} + \frac{47029}{99901} a^{15} + \frac{329894}{1698317} a^{14} + \frac{406211}{1698317} a^{13} - \frac{815859}{1698317} a^{12} - \frac{733206}{1698317} a^{11} + \frac{6995}{99901} a^{10} - \frac{26168}{1698317} a^{9} + \frac{221114}{1698317} a^{8} - \frac{710206}{1698317} a^{7} - \frac{255303}{1698317} a^{6} + \frac{609491}{1698317} a^{5} + \frac{23232}{99901} a^{4} + \frac{473596}{1698317} a^{3} - \frac{581665}{1698317} a^{2} + \frac{27777}{99901} a - \frac{53315}{1698317}$, $\frac{1}{1698317} a^{32} + \frac{18846}{1698317} a^{30} - \frac{12737}{1698317} a^{29} + \frac{38515}{1698317} a^{28} - \frac{26776}{1698317} a^{27} - \frac{235377}{1698317} a^{26} - \frac{264289}{1698317} a^{25} + \frac{279337}{1698317} a^{24} - \frac{506949}{1698317} a^{23} + \frac{13157}{99901} a^{22} - \frac{259476}{1698317} a^{21} + \frac{222282}{1698317} a^{20} - \frac{600387}{1698317} a^{19} - \frac{160791}{1698317} a^{18} + \frac{698389}{1698317} a^{17} + \frac{563379}{1698317} a^{16} + \frac{22601}{99901} a^{15} - \frac{679061}{1698317} a^{14} - \frac{385872}{1698317} a^{13} - \frac{428696}{1698317} a^{12} + \frac{695115}{1698317} a^{11} - \frac{754495}{1698317} a^{10} - \frac{600994}{1698317} a^{9} + \frac{510616}{1698317} a^{8} + \frac{16522}{1698317} a^{7} + \frac{738596}{1698317} a^{6} - \frac{36592}{1698317} a^{5} + \frac{381070}{1698317} a^{4} - \frac{839828}{1698317} a^{3} + \frac{673411}{1698317} a^{2} + \frac{24500}{99901} a - \frac{452537}{1698317}$, $\frac{1}{1698317} a^{33} + \frac{485}{99901} a^{30} - \frac{12609}{1698317} a^{29} + \frac{16634}{1698317} a^{28} + \frac{45739}{1698317} a^{27} - \frac{586905}{1698317} a^{26} - \frac{691180}{1698317} a^{25} + \frac{620195}{1698317} a^{24} + \frac{222290}{1698317} a^{23} + \frac{665749}{1698317} a^{22} - \frac{163735}{1698317} a^{21} - \frac{197190}{1698317} a^{20} + \frac{287248}{1698317} a^{19} - \frac{824528}{1698317} a^{18} + \frac{513564}{1698317} a^{17} + \frac{364402}{1698317} a^{16} - \frac{655517}{1698317} a^{15} - \frac{329362}{1698317} a^{14} + \frac{631339}{1698317} a^{13} - \frac{687794}{1698317} a^{12} + \frac{837580}{1698317} a^{11} + \frac{5455}{1698317} a^{10} + \frac{33024}{99901} a^{9} - \frac{127311}{1698317} a^{8} - \frac{454118}{1698317} a^{7} + \frac{371388}{1698317} a^{6} - \frac{68742}{1698317} a^{5} + \frac{568167}{1698317} a^{4} - \frac{460574}{1698317} a^{3} + \frac{138558}{1698317} a^{2} + \frac{776442}{1698317} a - \frac{529273}{1698317}$, $\frac{1}{1698317} a^{34} - \frac{20191}{1698317} a^{30} - \frac{21932}{1698317} a^{29} + \frac{21825}{1698317} a^{28} + \frac{19408}{1698317} a^{27} - \frac{103044}{1698317} a^{26} + \frac{152912}{1698317} a^{25} + \frac{317988}{1698317} a^{24} + \frac{301572}{1698317} a^{23} + \frac{551779}{1698317} a^{22} - \frac{74631}{1698317} a^{21} + \frac{696476}{1698317} a^{20} + \frac{453026}{1698317} a^{19} - \frac{603497}{1698317} a^{18} - \frac{399493}{1698317} a^{17} + \frac{1614}{1698317} a^{16} + \frac{417744}{1698317} a^{15} - \frac{37581}{99901} a^{14} - \frac{416761}{1698317} a^{13} - \frac{237810}{1698317} a^{12} + \frac{479217}{1698317} a^{11} + \frac{2126}{99901} a^{10} - \frac{158311}{1698317} a^{9} - \frac{645501}{1698317} a^{8} + \frac{622446}{1698317} a^{7} + \frac{190225}{1698317} a^{6} - \frac{464036}{1698317} a^{5} + \frac{598152}{1698317} a^{4} + \frac{70024}{1698317} a^{3} + \frac{657060}{1698317} a^{2} - \frac{51201}{1698317} a + \frac{217577}{1698317}$, $\frac{1}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{35} - \frac{474594083438788577194954238883048480110746348175531773913054208643932835437345100119619}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{34} + \frac{1226500729051593393751657965236925386999396788588698476840295511927771682296832439916917}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{33} - \frac{594261812283289236478941148651847650114165768489753456756003891946325936618751239385177}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{32} - \frac{69432076611289629736484501420097679449961900934422943612313778177779996445240296776434}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{31} + \frac{65616175035781091138800497639688313883332550435413712357339808248072510467616250650849318331}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{30} + \frac{124906933593552594982158627999765248215579736310775550473134068098713383983343551037188858519}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{29} - \frac{18621375380083435045470626183805310156021771462969550770334348353482793037582565770398309037}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{28} + \frac{22099496873684136764889299376993495215788437753363951268839513002331891265471952195595576412}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{27} - \frac{334621654822832177134392338850617234686559814648146648752096564486055941610862068189616870392}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{26} + \frac{168683021997577932108307431577815287358627859945998758766468273144099567196924767743098701706}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{25} + \frac{876626814612002883848498841412897643873311677442868810527568692079732028735772706865183521206}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{24} - \frac{1242207482975891260134492868053618119151021974098089592288940406828906668770982529253580470649}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{23} - \frac{1200122200917912845890316302723304717162118310223147779533492509487968030976179571801730394536}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{22} + \frac{2231300368538423612745692338357199223207706516223472009456126515108310196845004575224346975897}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{21} - \frac{1000589413528147687783040386947144800969742421248354699960455980014795839363149125977728965721}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{20} + \frac{1960057827426557903467942794700887576522027802450317154334129834874582272596412112627119287020}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{19} + \frac{1805416100700143487298667564362920606665531450841671545146925385034360078022123108686238512599}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{18} + \frac{298391605093810593948782380116014813642021663190950257214096001481331721745207670526950398297}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{17} + \frac{737406543704903154553997240589764396967688822628157331177450584816143524216269921069735499597}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{16} + \frac{142145906722665021930708609270938327120467724488792470981426071017080511680654399990813880497}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{15} - \frac{1578250360751486079129775194195902727003402163137866191173767815672145944991086010254748374076}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{14} - \frac{2064438007804994802083468488792995649947490538181050334967106654619599737205452465528143378637}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{13} + \frac{1979362548037493539488711977256435482673580220177777113457899060171729890275362153446110426360}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{12} - \frac{1853246439364240468909246571227220035014237738196823612195734840711451796605108942854098639382}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{11} - \frac{1435670326169181497766707437737326509296382143417235243391703299570284589572760886391110049551}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{10} - \frac{3608026264645853656597102511532110328779358810117919503075319264210990262020220355565871638}{270106715607522387295716878131616037174192155445233208151744488492929446964840477398166307491} a^{9} + \frac{1515621431028838586494540448877976803458957434877733122593687755877410688647636888806127789147}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{8} - \frac{442349100668407789574562456419438531588906999389216484961437220400299527277080446509191955053}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{7} - \frac{2074641578988835161676820240041126885471124023145507331155900388004193849866764617359468965614}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{6} + \frac{2014569110238979105317309531967248527421751530089497941345868961129837799453037604027816500266}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{5} + \frac{1185208016289532516149898542744930682517513353033741561244435910940614434076479153767253858816}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{4} + \frac{894060595101720876656381693872479124869495943508876980336421244230808018656226915248704907767}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{3} - \frac{602201506981001610303164856058219415101148029272105357844371383159858047592617386907183549937}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a^{2} + \frac{1533796403279418284317128937311999266362602158811910862866051257718686927155764034533899995222}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347} a + \frac{572419182371460738443882884127227546598340504499802096709378370438456164431040170173179876930}{4591814165327880584027186928237472631961266642568964538579656304379800598402288115768827227347}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 20453357667044700000000000 \) (assuming GRH)
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 4.4.19773.1, 6.6.14414517.1, \(\Q(\zeta_{27})^+\), 12.12.4108400332687853397.1, 18.18.10443002414754749649962321483613.1

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 $36$ R $36$ $36$ $36$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ $36$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $36$ $18^{2}$ $36$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $36$

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
3Data not computed
13Data not computed