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Properties

Label 25.25.1795754272...0625.2
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 151^{24}$
Root discriminant $1622.44$
Ramified primes $5, 151$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_{25}$ (as 25T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29654833162964959232, -913727091195101839360, 2164878525682671943680, -1078220626258959728640, -462138311426913280000, 350695231837169327104, 32320580026255119360, -34528266093143880000, -1138986477909910400, 1664070940159066800, 23546657206929408, -46443165424622140, -303022226928760, 811235682491605, 2451791469600, -9187449588510, -12107578640, 68008875655, 33413280, -323623200, -39864, 942995, 0, -1510, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1510*x^23 + 942995*x^21 - 39864*x^20 - 323623200*x^19 + 33413280*x^18 + 68008875655*x^17 - 12107578640*x^16 - 9187449588510*x^15 + 2451791469600*x^14 + 811235682491605*x^13 - 303022226928760*x^12 - 46443165424622140*x^11 + 23546657206929408*x^10 + 1664070940159066800*x^9 - 1138986477909910400*x^8 - 34528266093143880000*x^7 + 32320580026255119360*x^6 + 350695231837169327104*x^5 - 462138311426913280000*x^4 - 1078220626258959728640*x^3 + 2164878525682671943680*x^2 - 913727091195101839360*x - 29654833162964959232)
 
gp: K = bnfinit(x^25 - 1510*x^23 + 942995*x^21 - 39864*x^20 - 323623200*x^19 + 33413280*x^18 + 68008875655*x^17 - 12107578640*x^16 - 9187449588510*x^15 + 2451791469600*x^14 + 811235682491605*x^13 - 303022226928760*x^12 - 46443165424622140*x^11 + 23546657206929408*x^10 + 1664070940159066800*x^9 - 1138986477909910400*x^8 - 34528266093143880000*x^7 + 32320580026255119360*x^6 + 350695231837169327104*x^5 - 462138311426913280000*x^4 - 1078220626258959728640*x^3 + 2164878525682671943680*x^2 - 913727091195101839360*x - 29654833162964959232, 1)
 

Normalized defining polynomial

\( x^{25} - 1510 x^{23} + 942995 x^{21} - 39864 x^{20} - 323623200 x^{19} + 33413280 x^{18} + 68008875655 x^{17} - 12107578640 x^{16} - 9187449588510 x^{15} + 2451791469600 x^{14} + 811235682491605 x^{13} - 303022226928760 x^{12} - 46443165424622140 x^{11} + 23546657206929408 x^{10} + 1664070940159066800 x^{9} - 1138986477909910400 x^{8} - 34528266093143880000 x^{7} + 32320580026255119360 x^{6} + 350695231837169327104 x^{5} - 462138311426913280000 x^{4} - 1078220626258959728640 x^{3} + 2164878525682671943680 x^{2} - 913727091195101839360 x - 29654833162964959232 \)

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

Invariants

Degree:  $25$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[25, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(179575427285891034426038718238151945439156258996386904982500709593296051025390625=5^{40}\cdot 151^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1622.44$
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:  $5, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3775=5^{2}\cdot 151\)
Dirichlet character group:    $\lbrace$$\chi_{3775}(3416,·)$, $\chi_{3775}(1,·)$, $\chi_{3775}(2626,·)$, $\chi_{3775}(3331,·)$, $\chi_{3775}(836,·)$, $\chi_{3775}(2241,·)$, $\chi_{3775}(521,·)$, $\chi_{3775}(2061,·)$, $\chi_{3775}(846,·)$, $\chi_{3775}(1681,·)$, $\chi_{3775}(531,·)$, $\chi_{3775}(1876,·)$, $\chi_{3775}(1431,·)$, $\chi_{3775}(2611,·)$, $\chi_{3775}(2726,·)$, $\chi_{3775}(1896,·)$, $\chi_{3775}(2541,·)$, $\chi_{3775}(1711,·)$, $\chi_{3775}(2866,·)$, $\chi_{3775}(1331,·)$, $\chi_{3775}(1076,·)$, $\chi_{3775}(3446,·)$, $\chi_{3775}(1016,·)$, $\chi_{3775}(1596,·)$, $\chi_{3775}(1086,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} + \frac{3}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} - \frac{3}{128} a^{4} + \frac{3}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{11} - \frac{1}{256} a^{9} - \frac{5}{256} a^{7} - \frac{7}{256} a^{5} + \frac{3}{64} a^{3}$, $\frac{1}{2048} a^{12} + \frac{1}{1024} a^{11} + \frac{1}{2048} a^{10} - \frac{3}{1024} a^{9} + \frac{7}{2048} a^{8} - \frac{9}{1024} a^{7} - \frac{53}{2048} a^{6} + \frac{23}{1024} a^{5} - \frac{13}{512} a^{4} + \frac{61}{256} a^{3} + \frac{3}{64} a^{2} - \frac{1}{4} a$, $\frac{1}{4096} a^{13} - \frac{3}{4096} a^{11} - \frac{1}{512} a^{10} - \frac{13}{4096} a^{9} - \frac{81}{4096} a^{7} + \frac{11}{512} a^{6} - \frac{3}{256} a^{5} + \frac{15}{256} a^{4} + \frac{41}{256} a^{3} - \frac{5}{64} a^{2} - \frac{1}{8} a$, $\frac{1}{8192} a^{14} - \frac{1}{8192} a^{13} + \frac{1}{8192} a^{12} - \frac{13}{8192} a^{11} - \frac{1}{8192} a^{10} - \frac{27}{8192} a^{9} + \frac{11}{8192} a^{8} + \frac{113}{8192} a^{7} - \frac{55}{2048} a^{6} - \frac{59}{1024} a^{5} + \frac{17}{512} a^{4} + \frac{121}{512} a^{3} - \frac{1}{128} a^{2} - \frac{3}{16} a$, $\frac{1}{8192} a^{15} + \frac{5}{4096} a^{11} - \frac{1}{512} a^{10} - \frac{3}{1024} a^{9} + \frac{1}{512} a^{8} + \frac{61}{8192} a^{7} - \frac{11}{512} a^{6} + \frac{13}{256} a^{5} - \frac{9}{512} a^{4} - \frac{125}{512} a^{3} + \frac{5}{128} a^{2} + \frac{3}{16} a$, $\frac{1}{32768} a^{16} - \frac{1}{8192} a^{13} + \frac{1}{16384} a^{12} + \frac{11}{8192} a^{11} + \frac{13}{8192} a^{9} - \frac{123}{32768} a^{8} - \frac{135}{8192} a^{7} + \frac{29}{4096} a^{6} - \frac{3}{128} a^{5} - \frac{39}{2048} a^{4} - \frac{109}{512} a^{3} + \frac{1}{64} a^{2} + \frac{1}{4} a$, $\frac{1}{65536} a^{17} - \frac{1}{16384} a^{15} - \frac{1}{32768} a^{13} - \frac{15}{16384} a^{11} - \frac{1}{1024} a^{10} - \frac{103}{65536} a^{9} + \frac{1}{512} a^{8} - \frac{77}{8192} a^{7} - \frac{1}{1024} a^{6} + \frac{177}{4096} a^{5} + \frac{3}{64} a^{4} - \frac{1}{32} a^{3} - \frac{3}{64} a^{2}$, $\frac{1}{131072} a^{18} - \frac{1}{131072} a^{17} - \frac{1}{65536} a^{16} + \frac{1}{32768} a^{15} + \frac{3}{65536} a^{14} - \frac{7}{65536} a^{13} - \frac{1}{8192} a^{12} - \frac{53}{32768} a^{11} + \frac{241}{131072} a^{10} - \frac{73}{131072} a^{9} - \frac{83}{65536} a^{8} + \frac{71}{16384} a^{7} + \frac{5}{4096} a^{6} - \frac{129}{8192} a^{5} + \frac{41}{4096} a^{4} - \frac{9}{512} a^{3} - \frac{3}{256} a^{2} - \frac{15}{32} a - \frac{1}{2}$, $\frac{1}{262144} a^{19} - \frac{1}{262144} a^{18} - \frac{1}{131072} a^{15} + \frac{1}{131072} a^{14} + \frac{3}{65536} a^{13} - \frac{3}{65536} a^{12} + \frac{25}{262144} a^{11} + \frac{231}{262144} a^{10} + \frac{251}{65536} a^{9} - \frac{59}{65536} a^{8} - \frac{261}{16384} a^{7} - \frac{331}{16384} a^{6} - \frac{47}{4096} a^{5} + \frac{35}{4096} a^{4} + \frac{23}{128} a^{3} + \frac{3}{256} a^{2} + \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{1048576} a^{20} - \frac{3}{1048576} a^{18} - \frac{1}{524288} a^{17} + \frac{5}{524288} a^{16} + \frac{1}{131072} a^{15} - \frac{15}{524288} a^{14} + \frac{25}{262144} a^{13} - \frac{227}{1048576} a^{12} + \frac{251}{131072} a^{11} - \frac{751}{1048576} a^{10} + \frac{1655}{524288} a^{9} + \frac{59}{131072} a^{8} - \frac{1745}{65536} a^{7} - \frac{623}{65536} a^{6} - \frac{625}{32768} a^{5} + \frac{57}{4096} a^{4} - \frac{301}{2048} a^{3} - \frac{1}{256} a^{2} + \frac{3}{16} a$, $\frac{1}{247463936} a^{21} + \frac{1}{3866624} a^{20} - \frac{443}{247463936} a^{19} - \frac{7}{2097152} a^{18} - \frac{3}{123731968} a^{17} - \frac{3}{30932992} a^{16} + \frac{2057}{123731968} a^{15} + \frac{2421}{61865984} a^{14} + \frac{7709}{247463936} a^{13} + \frac{6023}{30932992} a^{12} + \frac{331865}{247463936} a^{11} + \frac{154043}{123731968} a^{10} - \frac{23515}{30932992} a^{9} + \frac{11731}{15466496} a^{8} + \frac{356097}{15466496} a^{7} - \frac{9325}{7733248} a^{6} - \frac{7655}{483328} a^{5} + \frac{17743}{483328} a^{4} - \frac{363}{7552} a^{3} - \frac{669}{7552} a^{2} - \frac{37}{472} a + \frac{10}{59}$, $\frac{1}{494927872} a^{22} + \frac{181}{494927872} a^{20} + \frac{75}{247463936} a^{19} - \frac{3}{247463936} a^{18} - \frac{427}{61865984} a^{17} + \frac{1881}{247463936} a^{16} - \frac{627}{123731968} a^{15} + \frac{3421}{494927872} a^{14} - \frac{5145}{61865984} a^{13} - \frac{20919}{494927872} a^{12} - \frac{363421}{247463936} a^{11} + \frac{72397}{61865984} a^{10} + \frac{7123}{30932992} a^{9} - \frac{60983}{30932992} a^{8} + \frac{183643}{15466496} a^{7} - \frac{20859}{966656} a^{6} + \frac{15603}{966656} a^{5} + \frac{2295}{241664} a^{4} + \frac{1205}{15104} a^{3} - \frac{2557}{15104} a^{2} - \frac{1}{1888} a + \frac{9}{118}$, $\frac{1}{3591196639232} a^{23} - \frac{2327}{3591196639232} a^{22} - \frac{103}{60867739648} a^{21} - \frac{1176493}{3591196639232} a^{20} - \frac{1632653}{1795598319616} a^{19} + \frac{6740595}{1795598319616} a^{18} - \frac{989149}{1795598319616} a^{17} - \frac{2362381}{1795598319616} a^{16} - \frac{192616339}{3591196639232} a^{15} - \frac{29654859}{3591196639232} a^{14} + \frac{217533655}{3591196639232} a^{13} - \frac{102241721}{3591196639232} a^{12} + \frac{461332807}{897799159808} a^{11} + \frac{961048203}{897799159808} a^{10} + \frac{48091759}{224449789952} a^{9} - \frac{876231495}{224449789952} a^{8} + \frac{1112511517}{56112447488} a^{7} + \frac{1443461997}{56112447488} a^{6} + \frac{235968121}{7014055936} a^{5} - \frac{128245563}{3507027968} a^{4} - \frac{45522659}{219189248} a^{3} - \frac{12312119}{54797312} a^{2} - \frac{572817}{1712416} a - \frac{58567}{214052}$, $\frac{1}{348873870119796983557661454393612243130379981074889778483829108393834019525686357109223432197136089094002794184485449732591513172297372156398259193185283654781985882112} a^{24} - \frac{6525495885347543359315885094785636934953330691962061846695057585263437048574215965253353433817888249920991291485766545342824365848355541135627008313432227}{87218467529949245889415363598403060782594995268722444620957277098458504881421589277305858049284022273500698546121362433147878293074343039099564798296320913695496470528} a^{23} - \frac{114047480564574638734446944627473524172628725227762457629958390129599879745811717915796709280733020581673711383222953661296585141146617291121249507867314266965}{174436935059898491778830727196806121565189990537444889241914554196917009762843178554611716098568044547001397092242724866295756586148686078199129596592641827390992941056} a^{22} - \frac{17997214251370920486127922141357916697595963169238903018932863500582150280145627124417184826305838634952791926325812303032031633079625164185033255529302778195}{87218467529949245889415363598403060782594995268722444620957277098458504881421589277305858049284022273500698546121362433147878293074343039099564798296320913695496470528} a^{21} + \frac{24692854186240892406727208282444926009774536715029149191339340151760059443840355128266992897805180790774138306337092769414874408583141990153584298575445135667335}{348873870119796983557661454393612243130379981074889778483829108393834019525686357109223432197136089094002794184485449732591513172297372156398259193185283654781985882112} a^{20} - \frac{29715000041527312389797678818695839681029814904264666183853783709668897597822079615137601913347162078193076591240071439708816005220517053590447176145435936047931}{43609233764974622944707681799201530391297497634361222310478638549229252440710794638652929024642011136750349273060681216573939146537171519549782399148160456847748235264} a^{19} + \frac{14597731314887574126241951123630745148153156780394756269002364973458189948213612129909007040462541756309420020427625768473965012743543146443115813039914008692657}{43609233764974622944707681799201530391297497634361222310478638549229252440710794638652929024642011136750349273060681216573939146537171519549782399148160456847748235264} a^{18} - \frac{218679674980936596726001882148924904813049475777734494731200001564858659356461705806556570100636400930400392552755611834600626826838906996322031583459643079204195}{43609233764974622944707681799201530391297497634361222310478638549229252440710794638652929024642011136750349273060681216573939146537171519549782399148160456847748235264} a^{17} - \frac{4164904207569717453025082114639317684094859678262234896321605540407742564301590784832052558028577386769469477535849048059905241603482252165255681047789949132277265}{348873870119796983557661454393612243130379981074889778483829108393834019525686357109223432197136089094002794184485449732591513172297372156398259193185283654781985882112} a^{16} + \frac{391316264464698976687871802984670616924478077027960942725872697562485908694612445343658493859635217649863385425824485563605178121315153223791241078756863733114697}{87218467529949245889415363598403060782594995268722444620957277098458504881421589277305858049284022273500698546121362433147878293074343039099564798296320913695496470528} a^{15} + \frac{9034211525255454505968917002644499266131999092692506189476244496591825722395425183707492606203343861248255591532159667038961237387867599308494157088767583449777271}{174436935059898491778830727196806121565189990537444889241914554196917009762843178554611716098568044547001397092242724866295756586148686078199129596592641827390992941056} a^{14} - \frac{4543394176074206120403453790270993421942636902345206826796199887032381987834903316563522880118178376071244298659669128354668153898174129371221617735651503154544927}{87218467529949245889415363598403060782594995268722444620957277098458504881421589277305858049284022273500698546121362433147878293074343039099564798296320913695496470528} a^{13} + \frac{52204041715245129640841827534471711377017573161895672356806520225610946887617859865958906095576861025016330981275413073837337587364450989308994071936073552782660729}{348873870119796983557661454393612243130379981074889778483829108393834019525686357109223432197136089094002794184485449732591513172297372156398259193185283654781985882112} a^{12} - \frac{2228107523289131927642587572522488144136596275412586654569094242239301097740439027855783599207260061341230256064637810699665484502855098215801174717071873256884825}{5451154220621827868088460224900191298912187204295152788809829818653656555088849329831616128080251392093793659132585152071742393317146439943722799893520057105968529408} a^{11} + \frac{23718755411357299633313950233008623175167881051307743674254204016906436175833777246332880773946655082809356556975178700714258218620162309548344251851351004828623749}{87218467529949245889415363598403060782594995268722444620957277098458504881421589277305858049284022273500698546121362433147878293074343039099564798296320913695496470528} a^{10} + \frac{36776361433583059964045487183485458202595487596267142360397424429942279504598840144629487395221615981545213341103354206500177568287738787308232517965472258936070811}{10902308441243655736176920449800382597824374408590305577619659637307313110177698659663232256160502784187587318265170304143484786634292879887445599787040114211937058816} a^{9} + \frac{20522204406405732545168589133785323263321761452970012613772541235001795181620077348967298667643629129318322042633357527215773824863649086533308692857858897204736727}{21804616882487311472353840899600765195648748817180611155239319274614626220355397319326464512321005568375174636530340608286969573268585759774891199574080228423874117632} a^{8} + \frac{5557137931109889500262347313248553813018054072883376505673100876505371411787985691290999751484284035776280508708219177961363473091658474081242020541494571882657973}{1362788555155456967022115056225047824728046801073788197202457454663414138772212332457904032020062848023448414783146288017935598329286609985930699973380014276492132352} a^{7} - \frac{95351807841160912585485164544821299726472291204680331777457378855629913237732541458388285655663513397433185249368838185587687669493339731806889669228179600852571737}{5451154220621827868088460224900191298912187204295152788809829818653656555088849329831616128080251392093793659132585152071742393317146439943722799893520057105968529408} a^{6} - \frac{7730263318418670394103562667363032949286020646713821245260245177644313334538090708592979469720298152754287290025007446193627794585586895640943814909869888674385455}{681394277577728483511057528112523912364023400536894098601228727331707069386106166228952016010031424011724207391573144008967799164643304992965349986690007138246066176} a^{5} + \frac{2174447460983402775550079704978088498116781427942055165403953081696572116605116291241440420732213556775938492808139381377642403482049638687480199100365123033352391}{340697138788864241755528764056261956182011700268447049300614363665853534693053083114476008005015712005862103695786572004483899582321652496482674993345003569123033088} a^{4} + \frac{3842475971172348415742941999147798802390479266640079648219714994177386649038413084669822726742735470008263054080224658892142472533995502259320200736242313289964949}{21293571174304015109720547753516372261375731266777940581288397729115845918315817694654750500313482000366381480986660750280243723895103281030167187084062723070189568} a^{3} - \frac{1283128136581821717943585260241912900807331470822544799296570559391215914914156191029340098828652581912701866451512033154616445128929700252643382206747534579348397}{5323392793576003777430136938379093065343932816694485145322099432278961479578954423663687625078370500091595370246665187570060930973775820257541796771015680767547392} a^{2} + \frac{1356026924895723495028769218941889119633443560698180329349949099608769013795494116408545424941408689995755114316542516357497070932446239985752240788315502771135}{83178012399625059022345889662173329145998950260851330395657803629358773118421162869745119141849539063931177660104143555782202046465247191524090574547120011992928} a + \frac{3241533783686781951831615626381045980999163904578260862252582705286807813626935648852890872483713276477346243653046553389687871603731515661876544227590494849577}{20794503099906264755586472415543332286499737565212832598914450907339693279605290717436279785462384765982794415026035888945550511616311797881022643636780002998232}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $24$
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:  \( 770271933701718500000000000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{25}$ (as 25T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 25
The 25 conjugacy class representatives for $C_{25}$
Character table for $C_{25}$ is not computed

Intermediate fields

5.5.519885601.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{25}$ $25$ R $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{25}$

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
5Data not computed
151Data not computed