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Properties

Label 36.36.9685961927...3568.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{99}\cdot 3^{48}\cdot 7^{24}$
Root discriminant $106.51$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (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![-3023, 134520, -1974252, 8023128, 40444434, -234531744, -523046756, 1490571564, 2779199601, -4344012424, -7580888508, 7325904048, 12401306684, -7951933872, -13272355962, 5873901604, 9770946912, -3042649560, -5094003336, 1121373960, 1910373432, -295082980, -518477394, 55109544, 101630814, -7185408, -14244732, 633928, 1399617, -35748, -93164, 1152, 3960, -16, -96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 96*x^34 - 16*x^33 + 3960*x^32 + 1152*x^31 - 93164*x^30 - 35748*x^29 + 1399617*x^28 + 633928*x^27 - 14244732*x^26 - 7185408*x^25 + 101630814*x^24 + 55109544*x^23 - 518477394*x^22 - 295082980*x^21 + 1910373432*x^20 + 1121373960*x^19 - 5094003336*x^18 - 3042649560*x^17 + 9770946912*x^16 + 5873901604*x^15 - 13272355962*x^14 - 7951933872*x^13 + 12401306684*x^12 + 7325904048*x^11 - 7580888508*x^10 - 4344012424*x^9 + 2779199601*x^8 + 1490571564*x^7 - 523046756*x^6 - 234531744*x^5 + 40444434*x^4 + 8023128*x^3 - 1974252*x^2 + 134520*x - 3023)
 
gp: K = bnfinit(x^36 - 96*x^34 - 16*x^33 + 3960*x^32 + 1152*x^31 - 93164*x^30 - 35748*x^29 + 1399617*x^28 + 633928*x^27 - 14244732*x^26 - 7185408*x^25 + 101630814*x^24 + 55109544*x^23 - 518477394*x^22 - 295082980*x^21 + 1910373432*x^20 + 1121373960*x^19 - 5094003336*x^18 - 3042649560*x^17 + 9770946912*x^16 + 5873901604*x^15 - 13272355962*x^14 - 7951933872*x^13 + 12401306684*x^12 + 7325904048*x^11 - 7580888508*x^10 - 4344012424*x^9 + 2779199601*x^8 + 1490571564*x^7 - 523046756*x^6 - 234531744*x^5 + 40444434*x^4 + 8023128*x^3 - 1974252*x^2 + 134520*x - 3023, 1)
 

Normalized defining polynomial

\( x^{36} - 96 x^{34} - 16 x^{33} + 3960 x^{32} + 1152 x^{31} - 93164 x^{30} - 35748 x^{29} + 1399617 x^{28} + 633928 x^{27} - 14244732 x^{26} - 7185408 x^{25} + 101630814 x^{24} + 55109544 x^{23} - 518477394 x^{22} - 295082980 x^{21} + 1910373432 x^{20} + 1121373960 x^{19} - 5094003336 x^{18} - 3042649560 x^{17} + 9770946912 x^{16} + 5873901604 x^{15} - 13272355962 x^{14} - 7951933872 x^{13} + 12401306684 x^{12} + 7325904048 x^{11} - 7580888508 x^{10} - 4344012424 x^{9} + 2779199601 x^{8} + 1490571564 x^{7} - 523046756 x^{6} - 234531744 x^{5} + 40444434 x^{4} + 8023128 x^{3} - 1974252 x^{2} + 134520 x - 3023 \)

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:  \(9685961927111642346915876016581426876940970357243365413693185450802413568=2^{99}\cdot 3^{48}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.51$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1008=2^{4}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1008}(1,·)$, $\chi_{1008}(193,·)$, $\chi_{1008}(781,·)$, $\chi_{1008}(877,·)$, $\chi_{1008}(529,·)$, $\chi_{1008}(277,·)$, $\chi_{1008}(793,·)$, $\chi_{1008}(25,·)$, $\chi_{1008}(925,·)$, $\chi_{1008}(289,·)$, $\chi_{1008}(37,·)$, $\chi_{1008}(169,·)$, $\chi_{1008}(541,·)$, $\chi_{1008}(949,·)$, $\chi_{1008}(841,·)$, $\chi_{1008}(697,·)$, $\chi_{1008}(445,·)$, $\chi_{1008}(373,·)$, $\chi_{1008}(961,·)$, $\chi_{1008}(709,·)$, $\chi_{1008}(673,·)$, $\chi_{1008}(457,·)$, $\chi_{1008}(589,·)$, $\chi_{1008}(205,·)$, $\chi_{1008}(337,·)$, $\chi_{1008}(85,·)$, $\chi_{1008}(121,·)$, $\chi_{1008}(421,·)$, $\chi_{1008}(865,·)$, $\chi_{1008}(613,·)$, $\chi_{1008}(361,·)$, $\chi_{1008}(109,·)$, $\chi_{1008}(625,·)$, $\chi_{1008}(757,·)$, $\chi_{1008}(505,·)$, $\chi_{1008}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{8} a^{28} - \frac{1}{2} a^{26} - \frac{1}{4} a^{24} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} + \frac{1}{8}$, $\frac{1}{8} a^{29} - \frac{1}{2} a^{27} - \frac{1}{4} a^{25} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} + \frac{1}{8} a$, $\frac{1}{2032} a^{30} + \frac{31}{1016} a^{29} - \frac{83}{2032} a^{28} + \frac{65}{254} a^{27} + \frac{169}{1016} a^{26} - \frac{111}{508} a^{25} + \frac{255}{1016} a^{24} - \frac{89}{508} a^{23} + \frac{193}{508} a^{22} + \frac{107}{508} a^{21} + \frac{253}{508} a^{20} + \frac{50}{127} a^{19} + \frac{35}{127} a^{18} - \frac{62}{127} a^{17} - \frac{165}{1016} a^{16} - \frac{149}{508} a^{15} - \frac{47}{1016} a^{14} - \frac{13}{254} a^{13} + \frac{51}{254} a^{12} + \frac{91}{254} a^{11} - \frac{101}{508} a^{10} + \frac{31}{508} a^{9} - \frac{5}{127} a^{8} - \frac{81}{508} a^{7} + \frac{21}{254} a^{6} + \frac{65}{254} a^{5} + \frac{13}{254} a^{4} + \frac{5}{127} a^{3} + \frac{137}{2032} a^{2} + \frac{331}{1016} a - \frac{63}{2032}$, $\frac{1}{2032} a^{31} - \frac{117}{2032} a^{29} + \frac{39}{1016} a^{28} - \frac{203}{1016} a^{27} + \frac{119}{254} a^{26} + \frac{49}{1016} a^{25} - \frac{30}{127} a^{24} + \frac{123}{508} a^{23} + \frac{79}{508} a^{22} - \frac{31}{508} a^{21} - \frac{123}{254} a^{20} - \frac{17}{127} a^{19} + \frac{54}{127} a^{18} + \frac{107}{1016} a^{17} - \frac{57}{254} a^{16} + \frac{395}{1016} a^{15} + \frac{161}{508} a^{14} - \frac{16}{127} a^{13} - \frac{23}{254} a^{12} + \frac{45}{508} a^{11} + \frac{197}{508} a^{10} + \frac{45}{254} a^{9} - \frac{111}{508} a^{8} - \frac{4}{127} a^{7} + \frac{33}{254} a^{6} + \frac{47}{254} a^{5} - \frac{17}{127} a^{4} - \frac{759}{2032} a^{3} + \frac{37}{254} a^{2} - \frac{721}{2032} a + \frac{175}{1016}$, $\frac{1}{16256} a^{32} - \frac{1}{4064} a^{31} - \frac{1}{8128} a^{30} + \frac{141}{4064} a^{29} - \frac{611}{16256} a^{28} + \frac{431}{2032} a^{27} + \frac{331}{2032} a^{26} + \frac{1241}{4064} a^{25} + \frac{559}{8128} a^{24} + \frac{193}{508} a^{23} - \frac{507}{1016} a^{22} - \frac{1025}{4064} a^{21} + \frac{1563}{4064} a^{20} - \frac{239}{508} a^{19} + \frac{3147}{8128} a^{18} - \frac{481}{1016} a^{17} - \frac{861}{2032} a^{16} - \frac{125}{508} a^{15} - \frac{1233}{8128} a^{14} - \frac{15}{254} a^{13} + \frac{1799}{4064} a^{12} + \frac{1643}{4064} a^{11} + \frac{895}{4064} a^{10} + \frac{23}{2032} a^{9} - \frac{341}{1016} a^{8} - \frac{41}{4064} a^{7} + \frac{149}{1016} a^{6} + \frac{903}{2032} a^{5} + \frac{2129}{16256} a^{4} - \frac{1947}{4064} a^{3} - \frac{2219}{8128} a^{2} + \frac{1807}{4064} a - \frac{7121}{16256}$, $\frac{1}{16256} a^{33} - \frac{1}{8128} a^{31} - \frac{1}{4064} a^{30} - \frac{403}{16256} a^{29} - \frac{9}{4064} a^{28} + \frac{403}{2032} a^{27} + \frac{1017}{4064} a^{26} + \frac{3511}{8128} a^{25} - \frac{785}{2032} a^{24} - \frac{235}{1016} a^{23} - \frac{129}{4064} a^{22} + \frac{39}{4064} a^{21} + \frac{241}{1016} a^{20} - \frac{2613}{8128} a^{19} - \frac{743}{2032} a^{18} + \frac{135}{2032} a^{17} - \frac{11}{508} a^{16} - \frac{625}{8128} a^{15} + \frac{419}{2032} a^{14} + \frac{151}{4064} a^{13} - \frac{249}{4064} a^{12} + \frac{1771}{4064} a^{11} - \frac{843}{2032} a^{10} - \frac{211}{1016} a^{9} + \frac{1895}{4064} a^{8} + \frac{26}{127} a^{7} - \frac{1009}{2032} a^{6} + \frac{5937}{16256} a^{5} + \frac{395}{2032} a^{4} + \frac{2501}{8128} a^{3} - \frac{307}{4064} a^{2} - \frac{801}{16256} a - \frac{965}{4064}$, $\frac{1}{3182696776144304238927564065597312} a^{34} - \frac{67781145604300929920838533017}{3182696776144304238927564065597312} a^{33} - \frac{5006228630556410622068091101}{3182696776144304238927564065597312} a^{32} + \frac{322041197637128366035997777693}{1591348388072152119463782032798656} a^{31} - \frac{324635653848003086992788997777}{3182696776144304238927564065597312} a^{30} + \frac{118677608688525979587741781545275}{3182696776144304238927564065597312} a^{29} - \frac{29271840998246256904971189150763}{3182696776144304238927564065597312} a^{28} + \frac{344426321851722031323766143673385}{795674194036076059731891016399328} a^{27} + \frac{313579980846673861116953553937305}{1591348388072152119463782032798656} a^{26} + \frac{510228658709235580994521407994455}{1591348388072152119463782032798656} a^{25} - \frac{530194749280972551748338305053649}{1591348388072152119463782032798656} a^{24} + \frac{335920484787948711093648851679019}{795674194036076059731891016399328} a^{23} + \frac{14593421209220022156265726857183}{198918548509019014932972754099832} a^{22} - \frac{21408961315156625804823980125349}{99459274254509507466486377049916} a^{21} - \frac{138110290121173469088667685119327}{1591348388072152119463782032798656} a^{20} + \frac{473115994751182784200727565863393}{1591348388072152119463782032798656} a^{19} - \frac{273184873697007295008941890482001}{1591348388072152119463782032798656} a^{18} - \frac{17840578036211116357955168750541}{397837097018038029865945508199664} a^{17} - \frac{188884380767225306560848101197965}{1591348388072152119463782032798656} a^{16} - \frac{604223963364641871748718058620059}{1591348388072152119463782032798656} a^{15} + \frac{210858332609561696428817454027893}{1591348388072152119463782032798656} a^{14} + \frac{37617566121653517997056873505343}{99459274254509507466486377049916} a^{13} - \frac{279356611103257839206811878289393}{795674194036076059731891016399328} a^{12} - \frac{194936295854023742897626591392681}{397837097018038029865945508199664} a^{11} - \frac{40953784291723609887039749885907}{795674194036076059731891016399328} a^{10} - \frac{76066439084496864208289434602159}{795674194036076059731891016399328} a^{9} + \frac{281091100750840000161202786398685}{795674194036076059731891016399328} a^{8} + \frac{174044124030992389586365303121129}{795674194036076059731891016399328} a^{7} + \frac{1370048666837171266183648283808201}{3182696776144304238927564065597312} a^{6} + \frac{1139541462893198432328002766366311}{3182696776144304238927564065597312} a^{5} - \frac{928349821538535262736775272045593}{3182696776144304238927564065597312} a^{4} - \frac{529615797706974424141153863136929}{1591348388072152119463782032798656} a^{3} - \frac{969494909501820393161094704008955}{3182696776144304238927564065597312} a^{2} + \frac{504629346866068941999919290969873}{3182696776144304238927564065597312} a + \frac{958912133365781704189228612184679}{3182696776144304238927564065597312}$, $\frac{1}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{35} - \frac{22342225622675657026789599882174991373173453}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{34} - \frac{1280930096485755734836564446786326703256483953848046734409733046784843457}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{33} - \frac{1083711331201362821961016448276095190211973257321412954821098533279007631}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{32} - \frac{7947308198926033472637380271814964842202769892266523269239300122790168495}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{31} - \frac{9468198409629837421185164790921611115943058803486199885096890700877849849}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{30} - \frac{836128443430597971168961051679859663530585847214873325230382917239997782121}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{29} + \frac{3668074466102487530424249219863344702352522687272227518133351222353749102649}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{28} + \frac{26985903114623945637877715505949914326203006255483977873587477035046893409085}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{27} - \frac{31385474466369355526224855949726877669644637308783338725584498629848978608343}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{26} - \frac{718926647129290383774644749006563463944232844749994631221988263609617844303}{2343476092189280355459505843775530463876509252853774715485741392443634865026} a^{25} + \frac{3454386735269546913822192955497365186155322356110650399583986843109764487425}{9373904368757121421838023375102121855506037011415098861942965569774539460104} a^{24} - \frac{651764112382723128071729592398547995004715573831547456433059104732611515525}{9373904368757121421838023375102121855506037011415098861942965569774539460104} a^{23} - \frac{9043288091495861580433270877170430817625142337260139759700140098357009306583}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{22} - \frac{36194856710647179806629130177291766269877848515868215989927279420681721965865}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{21} - \frac{11807456019751665110503973962676539882796104755311027402741773724557828401879}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{20} - \frac{1639804763871314360784671727991316712226261247046163986504554976860992880123}{9373904368757121421838023375102121855506037011415098861942965569774539460104} a^{19} + \frac{2497367874481107685693307197045765997080533205917864708936692420504134351567}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{18} - \frac{30097230539429250595487214991007437193270589828026499418286017208176973708025}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{17} - \frac{11430899960294951203545549411766154558709096200186441283271164605175302046143}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{16} + \frac{10615023431140708547740941191359722932613026426542473260032388351887891167237}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{15} + \frac{12664544495881936131040792199090475957501986854301124294786004877968151620765}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{14} - \frac{158039850777724417183412280327561599145608645883490939504904659896900566033}{2343476092189280355459505843775530463876509252853774715485741392443634865026} a^{13} + \frac{14345041229634435863248362495429623760196139704252743918182484938991318376125}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{12} - \frac{1062635029473972926744732825466914276961368504947927741877513258253897658811}{9373904368757121421838023375102121855506037011415098861942965569774539460104} a^{11} - \frac{10087769491704247590337987884813608237612719094072278887597371588578780079075}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{10} - \frac{8760209974633057067444691455417441279504425976460536578153842741717151851255}{37495617475028485687352093500408487422024148045660395447771862279098157840416} a^{9} - \frac{7026878076359624773980767542498304461669194466173734870458315854998716130505}{18747808737514242843676046750204243711012074022830197723885931139549078920208} a^{8} + \frac{19080217232884383886249658926372047249818273664831962951561728027251046421665}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{7} - \frac{19441084975355886148638694697828863547081389795848207528281183079972751932869}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{6} - \frac{19004033248741661842108886938456987422542094525704316613774921769541703512143}{74991234950056971374704187000816974844048296091320790895543724558196315680832} a^{5} - \frac{445236872349819670590532605934148107436349726986154303916851886140303460839}{2343476092189280355459505843775530463876509252853774715485741392443634865026} a^{4} + \frac{73779732815038682631841424997511094659090828676428059360891208398853342334203}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{3} - \frac{14164039932280244060370834451586875786632681935327968247482044391265641445643}{149982469900113942749408374001633949688096592182641581791087449116392631361664} a^{2} - \frac{92973279170966838896651176950604504013023226154666164021755221765741198017}{295241082480539257380725145672507774976568094847719649195054033693686282208} a - \frac{27972484259232301317385854667131145836948810391874760946396501774265116295647}{74991234950056971374704187000816974844048296091320790895543724558196315680832}$

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:  \( 18065118890547987000000000 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 3.3.3969.2, \(\Q(\zeta_{16})^+\), 6.6.3359232.1, 6.6.8065516032.1, 6.6.1229312.1, 6.6.8065516032.2, 9.9.62523502209.1, 12.12.369768517790072832.1, 12.12.2131641921124729651986432.1, 12.12.49519263525896192.1, 12.12.2131641921124729651986432.2, 18.18.524682375772545974113841184768.1

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$2$2.12.33.375$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
2.12.33.375$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
2.12.33.375$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
3Data not computed
7Data not computed