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Properties

Label 34.0.59904282501...0887.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,7^{17}\cdot 103^{32}$
Root discriminant $207.48$
Ramified primes $7, 103$
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![979230575957, 2091136690489, 3595506197489, 4220053914843, 4142556928386, 3151920159812, 2035799074068, 1039780899050, 468154476301, 186697982352, 97049072248, 54319604001, 28634333672, 8011080659, -234872300, -1585099529, -21438919, 682089833, 436056568, -858810, -95589136, -27680834, 13134497, 7444315, -521041, -1116463, -70798, 100041, 15232, -6939, -969, 310, 42, -15, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 15*x^33 + 42*x^32 + 310*x^31 - 969*x^30 - 6939*x^29 + 15232*x^28 + 100041*x^27 - 70798*x^26 - 1116463*x^25 - 521041*x^24 + 7444315*x^23 + 13134497*x^22 - 27680834*x^21 - 95589136*x^20 - 858810*x^19 + 436056568*x^18 + 682089833*x^17 - 21438919*x^16 - 1585099529*x^15 - 234872300*x^14 + 8011080659*x^13 + 28634333672*x^12 + 54319604001*x^11 + 97049072248*x^10 + 186697982352*x^9 + 468154476301*x^8 + 1039780899050*x^7 + 2035799074068*x^6 + 3151920159812*x^5 + 4142556928386*x^4 + 4220053914843*x^3 + 3595506197489*x^2 + 2091136690489*x + 979230575957)
 
gp: K = bnfinit(x^34 - 15*x^33 + 42*x^32 + 310*x^31 - 969*x^30 - 6939*x^29 + 15232*x^28 + 100041*x^27 - 70798*x^26 - 1116463*x^25 - 521041*x^24 + 7444315*x^23 + 13134497*x^22 - 27680834*x^21 - 95589136*x^20 - 858810*x^19 + 436056568*x^18 + 682089833*x^17 - 21438919*x^16 - 1585099529*x^15 - 234872300*x^14 + 8011080659*x^13 + 28634333672*x^12 + 54319604001*x^11 + 97049072248*x^10 + 186697982352*x^9 + 468154476301*x^8 + 1039780899050*x^7 + 2035799074068*x^6 + 3151920159812*x^5 + 4142556928386*x^4 + 4220053914843*x^3 + 3595506197489*x^2 + 2091136690489*x + 979230575957, 1)
 

Normalized defining polynomial

\( x^{34} - 15 x^{33} + 42 x^{32} + 310 x^{31} - 969 x^{30} - 6939 x^{29} + 15232 x^{28} + 100041 x^{27} - 70798 x^{26} - 1116463 x^{25} - 521041 x^{24} + 7444315 x^{23} + 13134497 x^{22} - 27680834 x^{21} - 95589136 x^{20} - 858810 x^{19} + 436056568 x^{18} + 682089833 x^{17} - 21438919 x^{16} - 1585099529 x^{15} - 234872300 x^{14} + 8011080659 x^{13} + 28634333672 x^{12} + 54319604001 x^{11} + 97049072248 x^{10} + 186697982352 x^{9} + 468154476301 x^{8} + 1039780899050 x^{7} + 2035799074068 x^{6} + 3151920159812 x^{5} + 4142556928386 x^{4} + 4220053914843 x^{3} + 3595506197489 x^{2} + 2091136690489 x + 979230575957 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 17]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5990428250146206797226208178461483076757442380295949051077252945668688444970887=-\,7^{17}\cdot 103^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $207.48$
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:  $7, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(721=7\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{721}(512,·)$, $\chi_{721}(1,·)$, $\chi_{721}(8,·)$, $\chi_{721}(267,·)$, $\chi_{721}(524,·)$, $\chi_{721}(13,·)$, $\chi_{721}(272,·)$, $\chi_{721}(538,·)$, $\chi_{721}(545,·)$, $\chi_{721}(34,·)$, $\chi_{721}(421,·)$, $\chi_{721}(167,·)$, $\chi_{721}(169,·)$, $\chi_{721}(426,·)$, $\chi_{721}(435,·)$, $\chi_{721}(694,·)$, $\chi_{721}(442,·)$, $\chi_{721}(699,·)$, $\chi_{721}(64,·)$, $\chi_{721}(323,·)$, $\chi_{721}(652,·)$, $\chi_{721}(587,·)$, $\chi_{721}(76,·)$, $\chi_{721}(594,·)$, $\chi_{721}(596,·)$, $\chi_{721}(608,·)$, $\chi_{721}(484,·)$, $\chi_{721}(615,·)$, $\chi_{721}(104,·)$, $\chi_{721}(491,·)$, $\chi_{721}(111,·)$, $\chi_{721}(370,·)$, $\chi_{721}(631,·)$, $\chi_{721}(505,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{149} a^{29} + \frac{17}{149} a^{28} + \frac{60}{149} a^{27} - \frac{23}{149} a^{26} + \frac{68}{149} a^{25} + \frac{37}{149} a^{24} - \frac{73}{149} a^{23} - \frac{59}{149} a^{22} - \frac{45}{149} a^{21} + \frac{15}{149} a^{20} - \frac{42}{149} a^{19} - \frac{70}{149} a^{18} - \frac{52}{149} a^{17} - \frac{38}{149} a^{15} - \frac{12}{149} a^{14} + \frac{46}{149} a^{13} - \frac{2}{149} a^{12} - \frac{44}{149} a^{11} - \frac{59}{149} a^{10} + \frac{57}{149} a^{9} - \frac{57}{149} a^{8} - \frac{55}{149} a^{7} - \frac{71}{149} a^{6} + \frac{29}{149} a^{5} + \frac{40}{149} a^{4} + \frac{29}{149} a^{3} - \frac{5}{149} a^{2} + \frac{2}{149} a + \frac{14}{149}$, $\frac{1}{7003} a^{30} - \frac{2}{7003} a^{29} + \frac{2568}{7003} a^{28} - \frac{2355}{7003} a^{27} - \frac{91}{7003} a^{26} - \frac{2447}{7003} a^{25} + \frac{2949}{7003} a^{24} - \frac{1950}{7003} a^{23} + \frac{1076}{7003} a^{22} - \frac{1216}{7003} a^{21} - \frac{2413}{7003} a^{20} - \frac{3444}{7003} a^{19} + \frac{5}{149} a^{18} - \frac{2886}{7003} a^{17} - \frac{1528}{7003} a^{16} + \frac{69}{149} a^{15} + \frac{1168}{7003} a^{14} - \frac{1770}{7003} a^{13} - \frac{2986}{7003} a^{12} - \frac{2501}{7003} a^{11} + \frac{2966}{7003} a^{10} + \frac{201}{7003} a^{9} - \frac{3144}{7003} a^{8} + \frac{676}{7003} a^{7} - \frac{1155}{7003} a^{6} + \frac{2767}{7003} a^{5} - \frac{1178}{7003} a^{4} - \frac{556}{7003} a^{3} + \frac{3077}{7003} a^{2} - \frac{2557}{7003} a + \frac{926}{7003}$, $\frac{1}{7003} a^{31} - \frac{21}{7003} a^{29} + \frac{854}{7003} a^{28} + \frac{1168}{7003} a^{27} + \frac{802}{7003} a^{26} - \frac{2650}{7003} a^{25} - \frac{14}{149} a^{24} - \frac{3200}{7003} a^{23} - \frac{615}{7003} a^{22} - \frac{568}{7003} a^{21} + \frac{1976}{7003} a^{20} - \frac{3128}{7003} a^{19} + \frac{3459}{7003} a^{18} + \frac{1066}{7003} a^{17} + \frac{187}{7003} a^{16} + \frac{839}{7003} a^{15} - \frac{3429}{7003} a^{14} + \frac{618}{7003} a^{13} - \frac{3303}{7003} a^{12} - \frac{344}{7003} a^{11} - \frac{2421}{7003} a^{10} - \frac{3024}{7003} a^{9} + \frac{1673}{7003} a^{8} + \frac{2312}{7003} a^{7} + \frac{1914}{7003} a^{6} - \frac{579}{7003} a^{5} - \frac{1267}{7003} a^{4} - \frac{2970}{7003} a^{3} + \frac{2516}{7003} a^{2} - \frac{2355}{7003} a + \frac{677}{7003}$, $\frac{1}{56958710768372279} a^{32} - \frac{3030545101876}{56958710768372279} a^{31} + \frac{816053983047}{56958710768372279} a^{30} - \frac{160532794358302}{56958710768372279} a^{29} + \frac{19880263118474160}{56958710768372279} a^{28} - \frac{23001667732194876}{56958710768372279} a^{27} - \frac{14199841988753377}{56958710768372279} a^{26} + \frac{22497790501942110}{56958710768372279} a^{25} + \frac{23720691159454628}{56958710768372279} a^{24} - \frac{28122486116722288}{56958710768372279} a^{23} + \frac{12941567418655953}{56958710768372279} a^{22} - \frac{10172970971346697}{56958710768372279} a^{21} + \frac{15930379091745964}{56958710768372279} a^{20} + \frac{26811954556194522}{56958710768372279} a^{19} - \frac{10155685043364364}{56958710768372279} a^{18} - \frac{22088766147772432}{56958710768372279} a^{17} - \frac{19538476807460252}{56958710768372279} a^{16} - \frac{21636918757177613}{56958710768372279} a^{15} + \frac{541418812549876}{56958710768372279} a^{14} - \frac{13658748419736342}{56958710768372279} a^{13} - \frac{5249484520945282}{56958710768372279} a^{12} + \frac{16338540161300308}{56958710768372279} a^{11} + \frac{14084854063772645}{56958710768372279} a^{10} + \frac{4605268187260473}{56958710768372279} a^{9} - \frac{2895810169930077}{56958710768372279} a^{8} + \frac{20268811079105695}{56958710768372279} a^{7} + \frac{3835781760046109}{56958710768372279} a^{6} - \frac{7855075368965693}{56958710768372279} a^{5} - \frac{23977528690220344}{56958710768372279} a^{4} + \frac{1088919127889432}{56958710768372279} a^{3} - \frac{1430046298405940}{56958710768372279} a^{2} - \frac{4525575827747281}{56958710768372279} a + \frac{6581184229484227}{56958710768372279}$, $\frac{1}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{33} + \frac{15680588018468250551543349847333018442635334119114059568507274410338270937336672018912383081790934400907306508472798458512690093280322490901133672516688}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{32} + \frac{77990124799559824010046018251850529106803339349661068701439888476202457461142993064442172303132024253286467074839255737072951703206601924032717000712571706302673351}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{31} + \frac{156237418401684799980531451173468061820727937935512792740678343583372220603502231784474396523559441362817560001582713859214735455260818905420628727025413571219876998}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{30} - \frac{6335886801660573973301367142349028007002088741110064682865373117938360338833054138907233786050097690539851110163296274150393238214538806920377344903474909292573673203}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{29} + \frac{709362282202869656683480121826741198904142250825361737878314838388891264775618216248826220672593716996272275181995847956481058430877391638527823890519712727207055998648}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{28} + \frac{392896687328379726856581918619796923684534991208488419191021744002692688260319462234436019411066360176256951638518868479921004431927010173186673523053152543121617188658}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{27} + \frac{995697682822334520768620739518154721582746586865611083281072906757750579271149122992175789906504705307413217595628107330275686865002757733255390128956256397986648872028}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{26} + \frac{670933490443948328273112627120807424638785775137283631530815851713944690872170435902709169624263042501290828478214913359175076131991068764012952089713952024603036072591}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{25} - \frac{236486366048132026857001826517432669362355188422315362454951400785862785891202109462286010909423503083494295748594737801930920345840871472863621139786395408288350694273}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{24} - \frac{667719210369231753506026512390524765515339694151604216534565009119372710685960231474861363679111879842028012029638322551755521629026403013478762373139574316324814289107}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{23} + \frac{471250351306680656241381691156449607929076198869435431938980135186468206865698886646443878778115897533339818404390753552819702675113406301670755982952112429388430310370}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{22} + \frac{699056161046142423553877576404204078587741134891655932243137578613065125993922029618105311725741784109504590062135707799866239781153325900774476702989437795977979548729}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{21} + \frac{1254565386709425142321457760654062820146904864616544779598526068310258418141994703067246889570328193999831915780366430116721760232186826389800296646548115103699936641963}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{20} + \frac{1240517597978420951941479970044324105813699478317307177712036443746138438942866579104220067324526055667869896045237735653603257234608087181890887821141483230047099085190}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{19} - \frac{5341594125245079140566252060686029372662613689004249015806962879097475003775674881372818986881073073992315706831966118320182545250221060857995400583072981195048440436}{59691655794446019410398453579658011449216469179989282496569089577866256717000958144802687335486703418175087406987174832391419732677837095821758434836801322100227803863} a^{18} - \frac{567389157077689390655426938540486904514690560418561327732335823368031874147840804697959039627992309893959405942287015814431905068310188816194814982589638338608416542400}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{17} - \frac{797818993886966979526677733864513167614581628929896031469377690749429495795402098082560434082990033895035465700029051570538256436218666354312842408284444065475926586155}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{16} + \frac{848264262572892721643077041291043756847427690169013516677567708950712070200247342005735937481819378722912179095512420118033788773002595162695657531338600862420422237904}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{15} - \frac{770776464105725864616613677166770520817654943185383402476455473049574278533611674817748916817543595639967884765165370551564774324115132526133604312928658480602872189525}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{14} + \frac{1375168696478498628406745767585537604504864082059020080125274654988693579141826436910429811285141410481799092145991801455285306965495995835464164702471006795833794381100}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{13} + \frac{978639117932686486840124378083814961756104888842985564844993572629850780069886188328669929289390409969856472563677173556426951736619908848983913649039148531094297286973}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{12} + \frac{461351631669854905682891514768803839045874049686287447668996011032034770668830114408049590829917614310602545282102599949430502642447269133732692069995326505888765622743}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{11} + \frac{1270002124747741765900726290632988064500428682493543362339882879539851623410110385615939737677149504837064635681422418061437252620165831751896128673850740093745974211865}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{10} + \frac{594229309644601548730078148937437647843317027188828217531089535256080113420862216028782790164892614562855291329594708507074567079578230763347842858770417160673322890554}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{9} - \frac{951459539464217431648710487564813271993491981029236969598773983191278337028800869973995229044500370709163381370165893703266027595501987970357350469198092654377584110345}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{8} - \frac{474496808664378354951306063402676300504375369113945847087686989392796473861875042532371615728781609282446817276876562007450210795528464189038865668768694147449220745917}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{7} - \frac{1231980262517972257180460106035367527311380090198046263292584579786877095271619759576763216885155775172617147562144541152423004089161878425250959337119434666297148514422}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{6} + \frac{421716369531237390026927065353019561557759389969253182065107533277449011306222849102264432824007354094168855883242218525493793313666990805763150803498351053750429326439}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{5} - \frac{328305094101922519409695443787370820924190156091332208378969147820521556122930349109988629501913486205517359289266948026818425610895836834764834491305849550156075365243}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{4} - \frac{187531433320915305968206371340731584013664827229776370321344671026099849049517222137446861655073632966118930965889343951379284690715640042087887672431192335897547283333}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{3} - \frac{452681110197768812296522839669336453966409278702738557303514530364297864880217583934311979039654402386986034254370053125196923123041768290834259192037029079485172631235}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a^{2} - \frac{820719755531503280904292728211466784449748544659768135913827331034167001636276424156836573994310575203658109761942691952708195642933595463318038520059346614221999595404}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561} a + \frac{351630637579214248887194313172877187001840305196243561738964188679576027852636454962017120655424058257992164958379027723025473286838690146013605668622280566909856371166}{2805507822338962912288727318243926538113174051459496277338747210159714065699045032805726304767875060654229108128397217122396727435858343503622646437329662138710706781561}$

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:  $16$
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

\(\Q(\sqrt{-7}) \), 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 $17^{2}$ $34$ $34$ R $17^{2}$ $34$ $34$ $34$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $34$ $17^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{17}$ $17^{2}$ $34$

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
7Data not computed
103Data not computed