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Properties

Label 34.34.5798570869...0768.1
Degree $34$
Signature $[34, 0]$
Discriminant $2^{51}\cdot 103^{32}$
Root discriminant $221.81$
Ramified primes $2, 103$
Class number Not computed
Class group Not computed
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6000161, 1256606, 225732609, -277250970, -2935988292, 6214331862, 13593855127, -46581664442, -1545819170, 120596752358, -81257174209, -131119332316, 161569633090, 48794706502, -136910136159, 17153938770, 60043936550, -21813383146, -14311196761, 8488774632, 1691394969, -1790460620, -23336117, 228417852, -20426311, -18136342, 2862995, 885574, -188915, -25224, 6820, 370, -129, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 2*x^33 - 129*x^32 + 370*x^31 + 6820*x^30 - 25224*x^29 - 188915*x^28 + 885574*x^27 + 2862995*x^26 - 18136342*x^25 - 20426311*x^24 + 228417852*x^23 - 23336117*x^22 - 1790460620*x^21 + 1691394969*x^20 + 8488774632*x^19 - 14311196761*x^18 - 21813383146*x^17 + 60043936550*x^16 + 17153938770*x^15 - 136910136159*x^14 + 48794706502*x^13 + 161569633090*x^12 - 131119332316*x^11 - 81257174209*x^10 + 120596752358*x^9 - 1545819170*x^8 - 46581664442*x^7 + 13593855127*x^6 + 6214331862*x^5 - 2935988292*x^4 - 277250970*x^3 + 225732609*x^2 + 1256606*x - 6000161)
 
gp: K = bnfinit(x^34 - 2*x^33 - 129*x^32 + 370*x^31 + 6820*x^30 - 25224*x^29 - 188915*x^28 + 885574*x^27 + 2862995*x^26 - 18136342*x^25 - 20426311*x^24 + 228417852*x^23 - 23336117*x^22 - 1790460620*x^21 + 1691394969*x^20 + 8488774632*x^19 - 14311196761*x^18 - 21813383146*x^17 + 60043936550*x^16 + 17153938770*x^15 - 136910136159*x^14 + 48794706502*x^13 + 161569633090*x^12 - 131119332316*x^11 - 81257174209*x^10 + 120596752358*x^9 - 1545819170*x^8 - 46581664442*x^7 + 13593855127*x^6 + 6214331862*x^5 - 2935988292*x^4 - 277250970*x^3 + 225732609*x^2 + 1256606*x - 6000161, 1)
 

Normalized defining polynomial

\( x^{34} - 2 x^{33} - 129 x^{32} + 370 x^{31} + 6820 x^{30} - 25224 x^{29} - 188915 x^{28} + 885574 x^{27} + 2862995 x^{26} - 18136342 x^{25} - 20426311 x^{24} + 228417852 x^{23} - 23336117 x^{22} - 1790460620 x^{21} + 1691394969 x^{20} + 8488774632 x^{19} - 14311196761 x^{18} - 21813383146 x^{17} + 60043936550 x^{16} + 17153938770 x^{15} - 136910136159 x^{14} + 48794706502 x^{13} + 161569633090 x^{12} - 131119332316 x^{11} - 81257174209 x^{10} + 120596752358 x^{9} - 1545819170 x^{8} - 46581664442 x^{7} + 13593855127 x^{6} + 6214331862 x^{5} - 2935988292 x^{4} - 277250970 x^{3} + 225732609 x^{2} + 1256606 x - 6000161 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57985708694758271640759625280476063462734635205978786696385026880737970925600768=2^{51}\cdot 103^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.81$
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:  $2, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(824=2^{3}\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{824}(1,·)$, $\chi_{824}(473,·)$, $\chi_{824}(133,·)$, $\chi_{824}(385,·)$, $\chi_{824}(9,·)$, $\chi_{824}(13,·)$, $\chi_{824}(529,·)$, $\chi_{824}(793,·)$, $\chi_{824}(409,·)$, $\chi_{824}(797,·)$, $\chi_{824}(545,·)$, $\chi_{824}(549,·)$, $\chi_{824}(169,·)$, $\chi_{824}(413,·)$, $\chi_{824}(821,·)$, $\chi_{824}(137,·)$, $\chi_{824}(641,·)$, $\chi_{824}(697,·)$, $\chi_{824}(61,·)$, $\chi_{824}(117,·)$, $\chi_{824}(581,·)$, $\chi_{824}(81,·)$, $\chi_{824}(729,·)$, $\chi_{824}(93,·)$, $\chi_{824}(421,·)$, $\chi_{824}(229,·)$, $\chi_{824}(785,·)$, $\chi_{824}(493,·)$, $\chi_{824}(317,·)$, $\chi_{824}(285,·)$, $\chi_{824}(373,·)$, $\chi_{824}(425,·)$, $\chi_{824}(505,·)$, $\chi_{824}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{47} a^{26} + \frac{11}{47} a^{25} + \frac{4}{47} a^{24} - \frac{16}{47} a^{23} + \frac{19}{47} a^{22} - \frac{14}{47} a^{21} + \frac{23}{47} a^{20} - \frac{22}{47} a^{19} - \frac{3}{47} a^{18} - \frac{15}{47} a^{17} - \frac{22}{47} a^{16} + \frac{17}{47} a^{15} + \frac{4}{47} a^{14} - \frac{5}{47} a^{13} + \frac{3}{47} a^{12} - \frac{2}{47} a^{11} - \frac{8}{47} a^{10} - \frac{10}{47} a^{9} - \frac{5}{47} a^{8} + \frac{6}{47} a^{7} + \frac{23}{47} a^{6} - \frac{3}{47} a^{5} + \frac{4}{47} a^{4} + \frac{10}{47} a^{3} - \frac{11}{47} a^{2} - \frac{19}{47} a$, $\frac{1}{47} a^{27} - \frac{23}{47} a^{25} - \frac{13}{47} a^{24} + \frac{7}{47} a^{23} + \frac{12}{47} a^{22} - \frac{11}{47} a^{21} + \frac{7}{47} a^{20} + \frac{4}{47} a^{19} + \frac{18}{47} a^{18} + \frac{2}{47} a^{17} - \frac{23}{47} a^{16} + \frac{5}{47} a^{15} - \frac{2}{47} a^{14} + \frac{11}{47} a^{13} + \frac{12}{47} a^{12} + \frac{14}{47} a^{11} - \frac{16}{47} a^{10} + \frac{11}{47} a^{9} + \frac{14}{47} a^{8} + \frac{4}{47} a^{7} - \frac{21}{47} a^{6} - \frac{10}{47} a^{5} + \frac{13}{47} a^{4} + \frac{20}{47} a^{3} + \frac{8}{47} a^{2} + \frac{21}{47} a$, $\frac{1}{47} a^{28} + \frac{5}{47} a^{25} + \frac{5}{47} a^{24} + \frac{20}{47} a^{23} + \frac{3}{47} a^{22} + \frac{14}{47} a^{21} + \frac{16}{47} a^{20} - \frac{18}{47} a^{19} - \frac{20}{47} a^{18} + \frac{8}{47} a^{17} + \frac{16}{47} a^{16} + \frac{13}{47} a^{15} + \frac{9}{47} a^{14} - \frac{9}{47} a^{13} - \frac{11}{47} a^{12} - \frac{15}{47} a^{11} + \frac{15}{47} a^{10} + \frac{19}{47} a^{9} - \frac{17}{47} a^{8} + \frac{23}{47} a^{7} + \frac{2}{47} a^{6} - \frac{9}{47} a^{5} + \frac{18}{47} a^{4} + \frac{3}{47} a^{3} + \frac{3}{47} a^{2} - \frac{14}{47} a$, $\frac{1}{47} a^{29} - \frac{3}{47} a^{25} - \frac{11}{47} a^{23} + \frac{13}{47} a^{22} - \frac{8}{47} a^{21} + \frac{8}{47} a^{20} - \frac{4}{47} a^{19} + \frac{23}{47} a^{18} - \frac{3}{47} a^{17} - \frac{18}{47} a^{16} + \frac{18}{47} a^{15} + \frac{18}{47} a^{14} + \frac{14}{47} a^{13} + \frac{17}{47} a^{12} - \frac{22}{47} a^{11} + \frac{12}{47} a^{10} - \frac{14}{47} a^{9} + \frac{1}{47} a^{8} + \frac{19}{47} a^{7} + \frac{17}{47} a^{6} - \frac{14}{47} a^{5} - \frac{17}{47} a^{4} - \frac{6}{47} a^{2} + \frac{1}{47} a$, $\frac{1}{7003} a^{30} + \frac{73}{7003} a^{29} - \frac{43}{7003} a^{28} + \frac{50}{7003} a^{27} + \frac{18}{7003} a^{26} - \frac{1494}{7003} a^{25} - \frac{2202}{7003} a^{24} - \frac{1589}{7003} a^{23} + \frac{1623}{7003} a^{22} + \frac{3336}{7003} a^{21} + \frac{2840}{7003} a^{20} - \frac{180}{7003} a^{19} - \frac{1562}{7003} a^{18} - \frac{2488}{7003} a^{17} + \frac{2232}{7003} a^{16} - \frac{2380}{7003} a^{15} - \frac{955}{7003} a^{14} - \frac{2077}{7003} a^{13} + \frac{287}{7003} a^{12} - \frac{1090}{7003} a^{11} + \frac{753}{7003} a^{10} + \frac{993}{7003} a^{9} + \frac{995}{7003} a^{8} - \frac{3301}{7003} a^{7} + \frac{1702}{7003} a^{6} + \frac{524}{7003} a^{5} - \frac{2127}{7003} a^{4} + \frac{1122}{7003} a^{3} - \frac{1666}{7003} a^{2} - \frac{1964}{7003} a - \frac{62}{149}$, $\frac{1}{7003} a^{31} - \frac{8}{7003} a^{29} + \frac{60}{7003} a^{28} - \frac{56}{7003} a^{27} + \frac{23}{7003} a^{26} + \frac{3007}{7003} a^{25} + \frac{3303}{7003} a^{24} - \frac{3219}{7003} a^{23} - \frac{115}{7003} a^{22} - \frac{202}{7003} a^{21} + \frac{1547}{7003} a^{20} - \frac{1534}{7003} a^{19} + \frac{3215}{7003} a^{18} + \frac{2374}{7003} a^{17} - \frac{1267}{7003} a^{16} + \frac{541}{7003} a^{15} + \frac{3}{149} a^{14} + \frac{226}{7003} a^{13} + \frac{905}{7003} a^{12} - \frac{2372}{7003} a^{11} + \frac{2644}{7003} a^{10} + \frac{1069}{7003} a^{9} - \frac{2479}{7003} a^{8} + \frac{2785}{7003} a^{7} + \frac{246}{7003} a^{6} + \frac{10}{47} a^{5} - \frac{3335}{7003} a^{4} - \frac{132}{7003} a^{3} - \frac{1483}{7003} a^{2} + \frac{845}{7003} a + \frac{56}{149}$, $\frac{1}{99752558263349} a^{32} - \frac{3482444919}{99752558263349} a^{31} - \frac{5990159047}{99752558263349} a^{30} - \frac{484602047993}{99752558263349} a^{29} + \frac{577467521980}{99752558263349} a^{28} + \frac{782843186446}{99752558263349} a^{27} - \frac{288257660973}{99752558263349} a^{26} + \frac{44310139075314}{99752558263349} a^{25} - \frac{28833517360776}{99752558263349} a^{24} - \frac{2428814492513}{99752558263349} a^{23} + \frac{28522523757575}{99752558263349} a^{22} - \frac{43032850563085}{99752558263349} a^{21} + \frac{39243726620737}{99752558263349} a^{20} + \frac{3550449236561}{99752558263349} a^{19} - \frac{16784517032239}{99752558263349} a^{18} - \frac{16219539142540}{99752558263349} a^{17} - \frac{16480994534991}{99752558263349} a^{16} - \frac{1163470818231}{99752558263349} a^{15} + \frac{24277185478329}{99752558263349} a^{14} - \frac{4873463063488}{99752558263349} a^{13} + \frac{1976415046923}{99752558263349} a^{12} - \frac{16941468255858}{99752558263349} a^{11} + \frac{25127342304138}{99752558263349} a^{10} + \frac{47493428785940}{99752558263349} a^{9} - \frac{12645517413582}{99752558263349} a^{8} + \frac{25950292870316}{99752558263349} a^{7} - \frac{28035665118226}{99752558263349} a^{6} + \frac{26780231021343}{99752558263349} a^{5} - \frac{38001084150309}{99752558263349} a^{4} - \frac{17957404958127}{99752558263349} a^{3} - \frac{39760342117340}{99752558263349} a^{2} + \frac{46531335752482}{99752558263349} a - \frac{337549893827}{2122394856667}$, $\frac{1}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{33} + \frac{8219684640641647258339351298123258221556836675878755453715031609101529285226063593864920829272941610830712361}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{32} + \frac{30681908480632513561116356788981959290345952530429901090715503929916392236523790118418721215381892028638737531962119360}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{31} - \frac{30053492792398040774996822646647279380582023879043255374684962639667596405344972408336556081601746696564034312950759381}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{30} + \frac{1072391372810199625235061360539537531847297504372322964984849014243857784728976320783962031796049475244775147013576639939}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{29} + \frac{19406508371477845307130040967729273714164632329343614119731807421650695053391647658421150596874681743967771440727913139426}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{28} + \frac{17335552595345209144280890741732212525548225165029966891713851477247123870629176299105443602957398146959258804620626251964}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{27} + \frac{4652351111815327393533825181513594139489636466246151774800205208286140071904222836141168236035398773920344702294333330757}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{26} + \frac{41113961087013430058133625316437407749601040455645879031418681715320228153538113163958473749025453042177479751903737989}{8114426975501355705229388348788426619742146124445927215164240981437550381115977049775626559826072110424233344834469151197} a^{25} - \frac{311717407181941739390078040651146352637331150043369372006915907183979066761679777206432283636321020463990825762739090252614}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{24} - \frac{272880439781480036418513889322030969328383251670600743646632163343154917044089910848481756813426038670349754548206721619497}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{23} + \frac{748429182657625010884024910439844041808762171925782283932295832630160630266535903250990745503933194111156656692345132003325}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{22} + \frac{1012669676509660524295898904266385467027104595793808976730908549501372393984649571382362817118063100451075782371858594114093}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{21} - \frac{132158560632155311230870479561756118852022451889068599524585024107044215240997266027132733816100030568959637965722332337475}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{20} - \frac{727774049847839641623047337846994768425037859480840622445290779105806921908619750778559369709813400734606531351328255560524}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{19} + \frac{14262483521927678339884710542807005461826865385861384762532850121731544953794133776887879792446445138540306511711119926991}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{18} + \frac{919926733157945562287132235379694763948376953163728437410667848483213874948537885627022784518332097622150018791013977567411}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{17} - \frac{612713066850199935563303225372024857732838668071976582252677676110648246852035764461243638282575932848303137366919935823720}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{16} - \frac{90651303516014559024387856074486155152009110643351171704731211136868421246058065270087886123450854249586046027998021587354}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{15} + \frac{524212040094430079468890422017902429822521575834246905988134925607870020065609100794508008015029789979498727766705736923487}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{14} - \frac{706369411906382612325807393645918851852223947323561281849240565566264102587361653358296506307052501050798876769061887538769}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{13} - \frac{141717070307083637209176472623347733476311557702608662046511903292268899800170281524040011422282700268751449055196318123128}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{12} - \frac{41373135320855078559434635310153938132771065326828570224707688087838753608287134153714915637824807988348367364383509464451}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{11} + \frac{120063868472786420345994889867634905681105305027506853450527212668347677085739396486742427687335853172292158675563093144518}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{10} + \frac{779779032642019351515447341162639662098430145877782597496273578579839242503588462671753430071516451588262171094823354584809}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{9} - \frac{974322529727785495038013975273440153174537134887014016796546974664794936479149531593168659730727709406258082496542072373040}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{8} - \frac{877440750491403714848130719170263839053585568825598116452387671129002930359277355088989106522393848966219569986358006659443}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{7} + \frac{114241503076425308135455600948820738231696366461538388506914181327811920943830553641176699551002422822071745590122514415}{45406261586316096818624024164496940446642221930410188459323305917405867026244722640233825217750148192373901482797135888613} a^{6} - \frac{625600130045619494102925155998557832095177810796810366826083775023964460843663848473998784549095716009234109423474692457239}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{5} + \frac{470352500664582268212029177953502743319773453540692237417889253424004548549765494969598298710173568214499656371914197403728}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{4} + \frac{456409687458410445933072029253275502594808931802093337836985605831846869898172400507451688581097676913224366119622449521465}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{3} + \frac{641295574887217713566939351839891933852294802977906585418651660739755136593799602018438376013650670751948852866981645667120}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a^{2} - \frac{986669795122398504416481699882045464380891324339852739358903126065734815052520656766549697810607876431573900979776675207909}{2134094294556856550475329135731356200992184430729278857588195378118075750233501964090989785234256965041573369691465386764811} a + \frac{5346076513999904848811148639106673936234824282335121747289716933416335387958439551556091202406377346536189164978076922820}{45406261586316096818624024164496940446642221930410188459323305917405867026244722640233825217750148192373901482797135888613}$

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:  $33$
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

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 17.17.160470643909878751793805444097921.1

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 $34$ $34$ $17^{2}$ $34$ $34$ $17^{2}$ $34$ $17^{2}$ $34$ $17^{2}$ $34$ $17^{2}$ $34$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{34}$ $34$ $34$

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
2Data not computed
103Data not computed