Properties

Label 35.35.732...081.1
Degree $35$
Signature $[35, 0]$
Discriminant $7.326\times 10^{79}$
Root discriminant $191.36$
Ramified primes $7, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{35}$ (as 35T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - 7*x^34 - 112*x^33 + 742*x^32 + 5845*x^31 - 34447*x^30 - 188041*x^29 + 918544*x^28 + 4120599*x^27 - 15488536*x^26 - 63930916*x^25 + 170889838*x^24 + 709660070*x^23 - 1224719069*x^22 - 5607885533*x^21 + 5331715046*x^20 + 31035849173*x^19 - 10542390278*x^18 - 116936498434*x^17 - 15812866636*x^16 + 286852619543*x^15 + 146466192098*x^14 - 426702558436*x^13 - 357822103121*x^12 + 338413935629*x^11 + 421000631443*x^10 - 94811511375*x^9 - 240823440823*x^8 - 25297046987*x^7 + 58988457696*x^6 + 13678049308*x^5 - 6371815744*x^4 - 1687698845*x^3 + 280068103*x^2 + 55092156*x - 2660503)
 
gp: K = bnfinit(x^35 - 7*x^34 - 112*x^33 + 742*x^32 + 5845*x^31 - 34447*x^30 - 188041*x^29 + 918544*x^28 + 4120599*x^27 - 15488536*x^26 - 63930916*x^25 + 170889838*x^24 + 709660070*x^23 - 1224719069*x^22 - 5607885533*x^21 + 5331715046*x^20 + 31035849173*x^19 - 10542390278*x^18 - 116936498434*x^17 - 15812866636*x^16 + 286852619543*x^15 + 146466192098*x^14 - 426702558436*x^13 - 357822103121*x^12 + 338413935629*x^11 + 421000631443*x^10 - 94811511375*x^9 - 240823440823*x^8 - 25297046987*x^7 + 58988457696*x^6 + 13678049308*x^5 - 6371815744*x^4 - 1687698845*x^3 + 280068103*x^2 + 55092156*x - 2660503, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2660503, 55092156, 280068103, -1687698845, -6371815744, 13678049308, 58988457696, -25297046987, -240823440823, -94811511375, 421000631443, 338413935629, -357822103121, -426702558436, 146466192098, 286852619543, -15812866636, -116936498434, -10542390278, 31035849173, 5331715046, -5607885533, -1224719069, 709660070, 170889838, -63930916, -15488536, 4120599, 918544, -188041, -34447, 5845, 742, -112, -7, 1]);
 

\( x^{35} - 7 x^{34} - 112 x^{33} + 742 x^{32} + 5845 x^{31} - 34447 x^{30} - 188041 x^{29} + 918544 x^{28} + 4120599 x^{27} - 15488536 x^{26} - 63930916 x^{25} + 170889838 x^{24} + 709660070 x^{23} - 1224719069 x^{22} - 5607885533 x^{21} + 5331715046 x^{20} + 31035849173 x^{19} - 10542390278 x^{18} - 116936498434 x^{17} - 15812866636 x^{16} + 286852619543 x^{15} + 146466192098 x^{14} - 426702558436 x^{13} - 357822103121 x^{12} + 338413935629 x^{11} + 421000631443 x^{10} - 94811511375 x^{9} - 240823440823 x^{8} - 25297046987 x^{7} + 58988457696 x^{6} + 13678049308 x^{5} - 6371815744 x^{4} - 1687698845 x^{3} + 280068103 x^{2} + 55092156 x - 2660503 \)

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

Invariants

Degree:  $35$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[35, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(732\!\cdots\!081\)\(\medspace = 7^{60}\cdot 11^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $191.36$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $35$
This field is Galois and abelian over $\Q$.
Conductor:  \(539=7^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{539}(1,·)$, $\chi_{539}(386,·)$, $\chi_{539}(267,·)$, $\chi_{539}(141,·)$, $\chi_{539}(526,·)$, $\chi_{539}(15,·)$, $\chi_{539}(400,·)$, $\chi_{539}(148,·)$, $\chi_{539}(533,·)$, $\chi_{539}(155,·)$, $\chi_{539}(36,·)$, $\chi_{539}(421,·)$, $\chi_{539}(295,·)$, $\chi_{539}(169,·)$, $\chi_{539}(302,·)$, $\chi_{539}(309,·)$, $\chi_{539}(190,·)$, $\chi_{539}(64,·)$, $\chi_{539}(449,·)$, $\chi_{539}(323,·)$, $\chi_{539}(71,·)$, $\chi_{539}(456,·)$, $\chi_{539}(78,·)$, $\chi_{539}(463,·)$, $\chi_{539}(344,·)$, $\chi_{539}(218,·)$, $\chi_{539}(92,·)$, $\chi_{539}(477,·)$, $\chi_{539}(225,·)$, $\chi_{539}(232,·)$, $\chi_{539}(113,·)$, $\chi_{539}(498,·)$, $\chi_{539}(372,·)$, $\chi_{539}(246,·)$, $\chi_{539}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{589} a^{30} + \frac{11}{589} a^{29} - \frac{157}{589} a^{28} + \frac{138}{589} a^{27} - \frac{135}{589} a^{26} - \frac{1}{19} a^{25} + \frac{56}{589} a^{24} - \frac{138}{589} a^{23} + \frac{208}{589} a^{22} + \frac{168}{589} a^{21} + \frac{115}{589} a^{20} - \frac{36}{589} a^{19} + \frac{104}{589} a^{18} + \frac{226}{589} a^{17} + \frac{28}{589} a^{16} - \frac{193}{589} a^{15} + \frac{195}{589} a^{14} - \frac{207}{589} a^{13} + \frac{128}{589} a^{12} - \frac{2}{31} a^{11} - \frac{60}{589} a^{10} - \frac{268}{589} a^{9} + \frac{243}{589} a^{8} - \frac{52}{589} a^{7} - \frac{101}{589} a^{6} + \frac{2}{31} a^{5} - \frac{257}{589} a^{4} - \frac{92}{589} a^{3} + \frac{65}{589} a^{2} - \frac{46}{589} a + \frac{127}{589}$, $\frac{1}{10378769} a^{31} + \frac{7766}{10378769} a^{30} + \frac{287175}{10378769} a^{29} + \frac{4123066}{10378769} a^{28} - \frac{1493862}{10378769} a^{27} - \frac{36821}{10378769} a^{26} + \frac{4044626}{10378769} a^{25} + \frac{3829138}{10378769} a^{24} + \frac{1852636}{10378769} a^{23} - \frac{383502}{10378769} a^{22} + \frac{1958512}{10378769} a^{21} - \frac{2954381}{10378769} a^{20} - \frac{3668771}{10378769} a^{19} + \frac{1682000}{10378769} a^{18} - \frac{4576147}{10378769} a^{17} + \frac{1259477}{10378769} a^{16} + \frac{525517}{10378769} a^{15} - \frac{1846460}{10378769} a^{14} - \frac{1234087}{10378769} a^{13} + \frac{4551340}{10378769} a^{12} - \frac{1856778}{10378769} a^{11} + \frac{24480}{10378769} a^{10} - \frac{2023320}{10378769} a^{9} - \frac{3661022}{10378769} a^{8} - \frac{3548032}{10378769} a^{7} - \frac{3344189}{10378769} a^{6} - \frac{209162}{10378769} a^{5} - \frac{2143322}{10378769} a^{4} - \frac{1264110}{10378769} a^{3} + \frac{31497}{334799} a^{2} - \frac{2378640}{10378769} a - \frac{56789}{154907}$, $\frac{1}{2044617493} a^{32} - \frac{39}{2044617493} a^{31} - \frac{39466}{107611447} a^{30} - \frac{326104149}{2044617493} a^{29} + \frac{695660712}{2044617493} a^{28} + \frac{960631140}{2044617493} a^{27} + \frac{146834605}{2044617493} a^{26} - \frac{759744149}{2044617493} a^{25} + \frac{249701760}{2044617493} a^{24} + \frac{961057362}{2044617493} a^{23} - \frac{689588551}{2044617493} a^{22} + \frac{261991073}{2044617493} a^{21} + \frac{690349314}{2044617493} a^{20} - \frac{991298293}{2044617493} a^{19} + \frac{390791271}{2044617493} a^{18} + \frac{505972028}{2044617493} a^{17} + \frac{162225098}{2044617493} a^{16} - \frac{521439281}{2044617493} a^{15} + \frac{527187975}{2044617493} a^{14} - \frac{21339268}{65955403} a^{13} - \frac{187947320}{2044617493} a^{12} + \frac{770104956}{2044617493} a^{11} - \frac{633297542}{2044617493} a^{10} - \frac{1738417}{2044617493} a^{9} - \frac{792077535}{2044617493} a^{8} + \frac{24516096}{2044617493} a^{7} - \frac{331089341}{2044617493} a^{6} + \frac{707994843}{2044617493} a^{5} + \frac{928375500}{2044617493} a^{4} - \frac{253002078}{2044617493} a^{3} - \frac{305004902}{2044617493} a^{2} + \frac{605681337}{2044617493} a - \frac{9708345}{30516679}$, $\frac{1}{762642324889} a^{33} + \frac{6}{24601365319} a^{32} + \frac{14005}{762642324889} a^{31} - \frac{552028720}{762642324889} a^{30} + \frac{139820372226}{762642324889} a^{29} - \frac{269456135987}{762642324889} a^{28} - \frac{95221432620}{762642324889} a^{27} + \frac{95749889222}{762642324889} a^{26} + \frac{17929278830}{762642324889} a^{25} - \frac{112411149238}{762642324889} a^{24} + \frac{186785099454}{762642324889} a^{23} + \frac{292566548450}{762642324889} a^{22} + \frac{62441330614}{762642324889} a^{21} + \frac{57982200458}{762642324889} a^{20} - \frac{240490313525}{762642324889} a^{19} + \frac{324567248727}{762642324889} a^{18} - \frac{8831150043}{40139069731} a^{17} + \frac{248589978890}{762642324889} a^{16} + \frac{337879496909}{762642324889} a^{15} + \frac{4542145212}{762642324889} a^{14} + \frac{257692187313}{762642324889} a^{13} + \frac{19517096351}{40139069731} a^{12} - \frac{340704426590}{762642324889} a^{11} - \frac{42287835181}{762642324889} a^{10} + \frac{95739530054}{762642324889} a^{9} + \frac{101151798298}{762642324889} a^{8} - \frac{235459904632}{762642324889} a^{7} + \frac{184862243720}{762642324889} a^{6} + \frac{34237460087}{762642324889} a^{5} + \frac{62374980640}{762642324889} a^{4} - \frac{61579045133}{762642324889} a^{3} - \frac{294156795291}{762642324889} a^{2} - \frac{239712546613}{762642324889} a - \frac{80678059}{11382721267}$, $\frac{1}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{34} + \frac{10282694668915148632341309013323281836573606136518431474742055273878839238966525274771496564740590908798082465533565877141679175}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{33} + \frac{33609178597524343649710063382524881413576037173597466583380614938742894381785107982713287245701955273807033957337984911683976243}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{32} - \frac{648913410817122617900405592407056132123416630673652035116866237520034060389081996334888116827931692224671556187339826358916713350419}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{31} + \frac{6939463028428174240724584116677551594802063819607294284458406999114294770895654767707726371584213264176139108768399268073498981764753437}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{30} + \frac{1546748416802518702812285978356791558404724618985619642882092895239383756310797931136054624316332712753747783712929166421627877597339887320}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{29} + \frac{7013127521991227867755339227293405743121747138480558669190700562132391932527102838159422600845503849522999452874192879570065735427445348889}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{28} - \frac{231222452876200104132604310542382083741737734098159101590668381271349185466865214089022010896622277504027992567643606479597999647994249272}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{27} + \frac{8141379893335166583786805365862982547021011218199307086009385846036079813795977863475027700590559471361952511889082308976620937268075082638}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{26} - \frac{2483713397598770093374724284152177303543004132952652767314204542102389822256345160126576220500249197866576380643119361024564035544693538695}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{25} - \frac{6164280962971621436630980632966107386659186132456019495336716398815153827141055962072086298314217663537986521242609543117493505111450237977}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{24} - \frac{4864539017790470709992877316701642435695699676624479624883363576632087621878057010816550967578825041683663072862445103888209061478520621637}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{23} - \frac{7249235305515182234203922006106510017871075373443440702527330094691682589173364991937513272304043473348405757953203383898248267650434723295}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{22} - \frac{2202339893438907033455626834455479682178072562408353418951328149478673313167804289731072062857893530472840829939280932763145446376193771260}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{21} + \frac{5570502201293668984079707755167614718017547950336117408545645520382622909606348989728351461747596979179315514834934235991685105787305524357}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{20} + \frac{1693102086579714840199992697454701408116230742745668946425815641042395236364259975354924679644241576589471179524546653180505328506316480892}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{19} - \frac{5964124731746852699356737236428573598098431672442836226545885229397230977309283359311860911518223933993866142775915101836704590864862320501}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{18} + \frac{22801631830946577046249982874577166101931732361968254088918428045000963352091146427653930097633765457886314626419798561279702334459820209}{84022586231008678230455423675385802805373454659307404678481042558685968893036619277650369626996309485813061639717272371007272135231264007} a^{17} + \frac{6636120914011755483968502758393646000268126852055097714841466177336037218170448875840292901994909140537085717093669886933740707669593097880}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{16} - \frac{1641131177735311897531915090761052195718994907970353715788121767985168546199254315559618618371319822333419422640143481639232219684862040342}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{15} + \frac{5885556340081650798885984353647842505011785458506311839921181965484403469631615048208913705145458660835338636672038832211141264014468359623}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{14} - \frac{7128079353302349973403525210153933354841678036975300929173583349668642869678484315435782714193388691652858272208373493607442204858292067157}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{13} - \frac{6139738962043459613001151834393310552948164298201595476070658167401304759168220648619060535151923454006698375863173046696055720713928291229}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{12} - \frac{7793199922667350910114930002037644858660078153599155041011393311876368970716583896777343252438379896701808729885660751194523299579825191830}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{11} - \frac{6516783287020941216858698845722152339366130608819848558274006319529515607084235406892798495898152588502324695085878815452543991038251654785}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{10} + \frac{7388145238039153023860049437859413203127216438321519795565754498546501247450781249152835119395156337193790782264999217771659324698638485806}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{9} - \frac{7688490411115185168361716254087590320462238542522775380610177106896332227285417297171701958126248548660293506576291869051158670291421776072}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{8} + \frac{2466584653725321731595741345987093400689604713035005662334992161110579361896492231581961853237652218223338291970283987571131387486456664013}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{7} + \frac{8224009514542576343771635868735839079039084635005208037577830536867280049237672277624282241823102801703953441608373349293850500271683858101}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{6} - \frac{3228704237155385004876667627428371954971679253847617028818269629626891857547321151429252976021678694097238202681572430368101984497022046730}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{5} + \frac{7960748516037942583682730453011850380355807561991291711768725991781215709479768501685234693479890735212478419437403587675941574252730106816}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{4} - \frac{3450190852131675322678976692974450250074590743612925909657107429967310461482649277643447478244432179214530571104114641365113954617947634217}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{3} - \frac{3591642428526003272816604601792267284134839682697249177930695140967725401723342959482363040976689367333856588307520157502590110146384540204}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a^{2} - \frac{75124870939962358978386046840090552461267791404465701970950077987012199882353621118270729394099618795616789648213462164000967716135741609}{16552449487508709611399718464051003152658570567883558721660765384061135871928213997697122816518272968705173143024302657088432610640559009379} a + \frac{59231880627696322212744015050147059893161981104307302089086339235097920345922094129729005824949356729889011261731067550050295198460093074}{247051484888189695692533111403746315711321948774381473457623363941210983163107671607419743530123477144853330492900039658036307621500880737}$

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

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $34$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 103376387230934820000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 103376387230934820000000000000 \cdot 1}{2\sqrt{73261800077965937220382205398471606200231960977600836588587975360331959691780081}}\approx 0.207492503370089$ (assuming GRH)

Galois group

$C_{35}$ (as 35T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 35
The 35 conjugacy class representatives for $C_{35}$
Character table for $C_{35}$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 7.7.13841287201.1

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 $35$ $35$ $35$ R R $35$ $35$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{5}$ $35$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{7}$ $35$ $35$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{5}$ $35$ $35$ $35$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
11Data not computed