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Properties

Label 36.36.3581730826...9357.1
Degree $36$
Signature $[36, 0]$
Discriminant $13^{27}\cdot 19^{34}$
Root discriminant $110.45$
Ramified primes $13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42979, 322424, -2246720, -10879867, 54837496, 82956251, -461721752, -323901451, 1967167029, 855286083, -4939927529, -1642297558, 7953185526, 2258507601, -8653274500, -2182363097, 6583639836, 1476195159, -3577294997, -701053767, 1403401302, 234115845, -398471838, -54785331, 81470952, 8895207, -11849973, -984922, 1201840, 72306, -82344, -3346, 3610, 88, -91, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 91*x^34 + 88*x^33 + 3610*x^32 - 3346*x^31 - 82344*x^30 + 72306*x^29 + 1201840*x^28 - 984922*x^27 - 11849973*x^26 + 8895207*x^25 + 81470952*x^24 - 54785331*x^23 - 398471838*x^22 + 234115845*x^21 + 1403401302*x^20 - 701053767*x^19 - 3577294997*x^18 + 1476195159*x^17 + 6583639836*x^16 - 2182363097*x^15 - 8653274500*x^14 + 2258507601*x^13 + 7953185526*x^12 - 1642297558*x^11 - 4939927529*x^10 + 855286083*x^9 + 1967167029*x^8 - 323901451*x^7 - 461721752*x^6 + 82956251*x^5 + 54837496*x^4 - 10879867*x^3 - 2246720*x^2 + 322424*x + 42979)
 
gp: K = bnfinit(x^36 - x^35 - 91*x^34 + 88*x^33 + 3610*x^32 - 3346*x^31 - 82344*x^30 + 72306*x^29 + 1201840*x^28 - 984922*x^27 - 11849973*x^26 + 8895207*x^25 + 81470952*x^24 - 54785331*x^23 - 398471838*x^22 + 234115845*x^21 + 1403401302*x^20 - 701053767*x^19 - 3577294997*x^18 + 1476195159*x^17 + 6583639836*x^16 - 2182363097*x^15 - 8653274500*x^14 + 2258507601*x^13 + 7953185526*x^12 - 1642297558*x^11 - 4939927529*x^10 + 855286083*x^9 + 1967167029*x^8 - 323901451*x^7 - 461721752*x^6 + 82956251*x^5 + 54837496*x^4 - 10879867*x^3 - 2246720*x^2 + 322424*x + 42979, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 91 x^{34} + 88 x^{33} + 3610 x^{32} - 3346 x^{31} - 82344 x^{30} + 72306 x^{29} + 1201840 x^{28} - 984922 x^{27} - 11849973 x^{26} + 8895207 x^{25} + 81470952 x^{24} - 54785331 x^{23} - 398471838 x^{22} + 234115845 x^{21} + 1403401302 x^{20} - 701053767 x^{19} - 3577294997 x^{18} + 1476195159 x^{17} + 6583639836 x^{16} - 2182363097 x^{15} - 8653274500 x^{14} + 2258507601 x^{13} + 7953185526 x^{12} - 1642297558 x^{11} - 4939927529 x^{10} + 855286083 x^{9} + 1967167029 x^{8} - 323901451 x^{7} - 461721752 x^{6} + 82956251 x^{5} + 54837496 x^{4} - 10879867 x^{3} - 2246720 x^{2} + 322424 x + 42979 \)

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:  \(35817308260975188347623162969257824484378877602003672125976639688418239357=13^{27}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.45$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(131,·)$, $\chi_{247}(135,·)$, $\chi_{247}(8,·)$, $\chi_{247}(66,·)$, $\chi_{247}(142,·)$, $\chi_{247}(144,·)$, $\chi_{247}(18,·)$, $\chi_{247}(148,·)$, $\chi_{247}(21,·)$, $\chi_{247}(151,·)$, $\chi_{247}(25,·)$, $\chi_{247}(157,·)$, $\chi_{247}(31,·)$, $\chi_{247}(34,·)$, $\chi_{247}(164,·)$, $\chi_{247}(168,·)$, $\chi_{247}(92,·)$, $\chi_{247}(174,·)$, $\chi_{247}(60,·)$, $\chi_{247}(64,·)$, $\chi_{247}(194,·)$, $\chi_{247}(196,·)$, $\chi_{247}(70,·)$, $\chi_{247}(200,·)$, $\chi_{247}(203,·)$, $\chi_{247}(77,·)$, $\chi_{247}(207,·)$, $\chi_{247}(86,·)$, $\chi_{247}(220,·)$, $\chi_{247}(233,·)$, $\chi_{247}(235,·)$, $\chi_{247}(109,·)$, $\chi_{247}(242,·)$, $\chi_{247}(118,·)$, $\chi_{247}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{1901} a^{27} + \frac{633}{1901} a^{26} - \frac{80}{1901} a^{25} + \frac{687}{1901} a^{24} - \frac{49}{1901} a^{23} - \frac{574}{1901} a^{22} - \frac{740}{1901} a^{21} + \frac{68}{1901} a^{20} - \frac{648}{1901} a^{19} + \frac{450}{1901} a^{18} + \frac{174}{1901} a^{17} + \frac{740}{1901} a^{16} - \frac{166}{1901} a^{15} + \frac{315}{1901} a^{14} + \frac{184}{1901} a^{13} + \frac{39}{1901} a^{12} + \frac{343}{1901} a^{11} - \frac{48}{1901} a^{10} + \frac{382}{1901} a^{9} + \frac{541}{1901} a^{8} - \frac{227}{1901} a^{7} - \frac{635}{1901} a^{6} - \frac{668}{1901} a^{5} + \frac{81}{1901} a^{4} - \frac{406}{1901} a^{3} - \frac{383}{1901} a^{2} - \frac{866}{1901} a + \frac{728}{1901}$, $\frac{1}{1901} a^{28} + \frac{342}{1901} a^{26} + \frac{409}{1901} a^{24} + \frac{27}{1901} a^{23} - \frac{489}{1901} a^{22} + \frac{842}{1901} a^{21} + \frac{31}{1901} a^{20} + \frac{18}{1901} a^{19} + \frac{474}{1901} a^{18} + \frac{856}{1901} a^{17} - \frac{940}{1901} a^{16} + \frac{838}{1901} a^{15} + \frac{394}{1901} a^{14} - \frac{472}{1901} a^{13} + \frac{369}{1901} a^{12} - \frac{453}{1901} a^{11} + \frac{350}{1901} a^{10} + \frac{162}{1901} a^{9} - \frac{500}{1901} a^{8} + \frac{481}{1901} a^{7} + \frac{176}{1901} a^{6} + \frac{903}{1901} a^{5} - \frac{352}{1901} a^{4} - \frac{20}{1901} a^{3} + \frac{146}{1901} a^{2} - \frac{483}{1901} a - \frac{782}{1901}$, $\frac{1}{1901} a^{29} + \frac{228}{1901} a^{26} - \frac{746}{1901} a^{25} + \frac{797}{1901} a^{24} - \frac{840}{1901} a^{23} - \frac{554}{1901} a^{22} + \frac{278}{1901} a^{21} - \frac{426}{1901} a^{20} - \frac{327}{1901} a^{19} + \frac{937}{1901} a^{18} + \frac{384}{1901} a^{17} + \frac{591}{1901} a^{16} + \frac{136}{1901} a^{15} + \frac{155}{1901} a^{14} + \frac{174}{1901} a^{13} - \frac{484}{1901} a^{12} + \frac{906}{1901} a^{11} - \frac{531}{1901} a^{10} + \frac{25}{1901} a^{9} - \frac{144}{1901} a^{8} - \frac{131}{1901} a^{7} - \frac{542}{1901} a^{6} - \frac{16}{1901} a^{5} + \frac{793}{1901} a^{4} + \frac{225}{1901} a^{3} - \frac{666}{1901} a^{2} + \frac{735}{1901} a + \frac{55}{1901}$, $\frac{1}{1901} a^{30} - \frac{594}{1901} a^{26} + \frac{27}{1901} a^{25} + \frac{307}{1901} a^{24} - \frac{788}{1901} a^{23} - \frac{19}{1901} a^{22} - \frac{895}{1901} a^{21} - \frac{623}{1901} a^{20} + \frac{403}{1901} a^{19} + \frac{438}{1901} a^{18} + \frac{840}{1901} a^{17} + \frac{605}{1901} a^{16} - \frac{17}{1901} a^{15} + \frac{592}{1901} a^{14} - \frac{614}{1901} a^{13} - \frac{382}{1901} a^{12} - \frac{794}{1901} a^{11} - \frac{437}{1901} a^{10} + \frac{206}{1901} a^{9} + \frac{86}{1901} a^{8} - \frac{113}{1901} a^{7} + \frac{288}{1901} a^{6} - \frac{884}{1901} a^{5} + \frac{767}{1901} a^{4} + \frac{654}{1901} a^{3} + \frac{613}{1901} a^{2} - \frac{201}{1901} a - \frac{597}{1901}$, $\frac{1}{1901} a^{31} - \frac{369}{1901} a^{26} + \frac{312}{1901} a^{25} + \frac{476}{1901} a^{24} - \frac{610}{1901} a^{23} + \frac{329}{1901} a^{22} + \frac{849}{1901} a^{21} + \frac{874}{1901} a^{20} - \frac{472}{1901} a^{19} + \frac{99}{1901} a^{18} - \frac{594}{1901} a^{17} + \frac{412}{1901} a^{16} + \frac{840}{1901} a^{15} + \frac{198}{1901} a^{14} + \frac{557}{1901} a^{13} - \frac{440}{1901} a^{12} - \frac{102}{1901} a^{11} + \frac{209}{1901} a^{10} + \frac{775}{1901} a^{9} - \frac{28}{1901} a^{8} + \frac{421}{1901} a^{7} + \frac{225}{1901} a^{6} - \frac{617}{1901} a^{5} - \frac{658}{1901} a^{4} + \frac{876}{1901} a^{3} + \frac{417}{1901} a^{2} + \frac{170}{1901} a + \frac{905}{1901}$, $\frac{1}{1901} a^{32} + \frac{66}{1901} a^{26} - \frac{529}{1901} a^{25} + \frac{60}{1901} a^{24} - \frac{643}{1901} a^{23} + \frac{54}{1901} a^{22} - \frac{343}{1901} a^{21} - \frac{93}{1901} a^{20} + \frac{513}{1901} a^{19} + \frac{69}{1901} a^{18} - \frac{16}{1901} a^{17} + \frac{156}{1901} a^{16} - \frac{224}{1901} a^{15} + \frac{831}{1901} a^{14} + \frac{921}{1901} a^{13} - \frac{919}{1901} a^{12} - \frac{591}{1901} a^{11} + \frac{172}{1901} a^{10} + \frac{256}{1901} a^{9} + \frac{445}{1901} a^{8} + \frac{106}{1901} a^{7} + \frac{792}{1901} a^{6} - \frac{20}{1901} a^{5} + \frac{349}{1901} a^{4} + \frac{782}{1901} a^{3} - \frac{483}{1901} a^{2} + \frac{719}{1901} a + \frac{591}{1901}$, $\frac{1}{363091} a^{33} - \frac{82}{363091} a^{32} + \frac{87}{363091} a^{31} - \frac{80}{363091} a^{30} + \frac{15}{363091} a^{29} + \frac{85}{363091} a^{28} - \frac{83}{363091} a^{27} + \frac{2778}{363091} a^{26} - \frac{22096}{363091} a^{25} - \frac{50057}{363091} a^{24} - \frac{75225}{363091} a^{23} - \frac{113869}{363091} a^{22} + \frac{180802}{363091} a^{21} - \frac{18643}{363091} a^{20} + \frac{150867}{363091} a^{19} + \frac{141492}{363091} a^{18} + \frac{47343}{363091} a^{17} + \frac{180937}{363091} a^{16} + \frac{104197}{363091} a^{15} + \frac{45506}{363091} a^{14} - \frac{9570}{363091} a^{13} + \frac{136668}{363091} a^{12} - \frac{107713}{363091} a^{11} - \frac{32521}{363091} a^{10} - \frac{139742}{363091} a^{9} + \frac{121785}{363091} a^{8} + \frac{38272}{363091} a^{7} + \frac{177492}{363091} a^{6} - \frac{4528}{363091} a^{5} + \frac{17366}{363091} a^{4} - \frac{136329}{363091} a^{3} - \frac{108752}{363091} a^{2} - \frac{110989}{363091} a - \frac{90381}{363091}$, $\frac{1}{363091} a^{34} + \frac{48}{363091} a^{32} - \frac{13}{363091} a^{31} - \frac{51}{363091} a^{30} - \frac{22}{363091} a^{29} + \frac{11}{363091} a^{28} - \frac{17}{363091} a^{27} + \frac{5914}{363091} a^{26} + \frac{147391}{363091} a^{25} - \frac{98802}{363091} a^{24} - \frac{116648}{363091} a^{23} - \frac{2781}{363091} a^{22} - \frac{76172}{363091} a^{21} + \frac{26182}{363091} a^{20} + \frac{2086}{363091} a^{19} - \frac{70264}{363091} a^{18} - \frac{80491}{363091} a^{17} - \frac{136891}{363091} a^{16} - \frac{117839}{363091} a^{15} - \frac{97123}{363091} a^{14} - \frac{116328}{363091} a^{13} - \frac{143388}{363091} a^{12} - \frac{174487}{363091} a^{11} + \frac{127296}{363091} a^{10} + \frac{54372}{363091} a^{9} - \frac{50608}{363091} a^{8} - \frac{96419}{363091} a^{7} + \frac{26749}{363091} a^{6} + \frac{157759}{363091} a^{5} + \frac{50003}{363091} a^{4} + \frac{8010}{363091} a^{3} - \frac{165107}{363091} a^{2} - \frac{38759}{363091} a - \frac{63090}{363091}$, $\frac{1}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{35} + \frac{332406546895359587056202944795211633796797338950789844085360172324436867791004390}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{34} + \frac{423658127194643906789514234152740340563252722559667448117801691588731914959461775}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{33} + \frac{85466862563593168019131174573356022969192633685475588248381248200087997996236750250}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{32} - \frac{6623588713078673900771244507501019458678536239061078027209997897820434210304771280}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{31} - \frac{58871183784064496856095742433662275298940745614759436178722679897117007770252242933}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{30} + \frac{42906757128466819771729271580032171801740527003282349096097177886235575542916078216}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{29} + \frac{24690217477860615447531306251882345560430767090151173313375278528927813468852643773}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{28} - \frac{7758965237306785369836708306713943137531004268538432691649470243137149236137402248}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{27} - \frac{35642174192901344416056360958038529115445498247632429648395275298569617281908170339313}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{26} - \frac{4080701825336305045212645051037242948782205714787255621057352317078226181894267406654}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{25} - \frac{82556576856810572439170262840464790846543922625748140925043153839017284829954143798440}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{24} + \frac{157630083249143668369839965695489016552496718941983874389812035669491000820539775220758}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{23} - \frac{90611465941644242517470956298115040145991160983820470589000782037923994445806742489871}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{22} - \frac{147057891861947255030175920791960564329015633694976852895101729457863538507545877090193}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{21} - \frac{207088700020335890504066268869250809948635812045527134395800110639874231084591229615385}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{20} - \frac{119807840414103063068319063368289000599609817282495016590632832746888091944164443577492}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{19} + \frac{47759106962741132150239653706610671582169482395573647434873059886817812508024730704741}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{18} - \frac{18213660001299063170649537611669927432522371193177020536908322339430849793220992044817}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{17} + \frac{23643202317683567863271926217367854333844964598864361135389780117151804630462032083745}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{16} + \frac{159740854172238636801416362235145152782255635421547615050614119233291749455237668554789}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{15} + \frac{124622540665698208371721681994404446952613618938013427545748357609415888425216381794375}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{14} + \frac{46646105540070416650251334337287304109489899860701215535858054995057620439803835156617}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{13} + \frac{153420465959506950135190152702739563834794877523091192423183951816130662045311258724379}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{12} - \frac{10748941369767594232215429844177738304781593842480941659298796646521559497869737975913}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{11} - \frac{224926167517459685331153721572473165231128508265921757211202911277981926575671379097730}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{10} - \frac{218903060082661981726004176519450634565293854513368300176884608360967682787262974762781}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{9} - \frac{41442223368890536705159633207870289925543290093779996915440145568018551383310449707798}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{8} - \frac{156106478606524490163924842568600819584760273597656198144397363545791621477490487682708}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{7} + \frac{52200417779231091164544705811433853181971326851509685295930693733961817785766738834954}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{6} + \frac{91781199374991509151609213183283185656825703970044568022936781969302388454560716527139}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{5} + \frac{130320392259220844141867641868399808126779307859196827372688354803513868477902952750507}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{4} + \frac{145102894356154083265903323088208141760875498307144971817090635436048060087497358928731}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{3} - \frac{145756969446626643491156239155171506860071292414864760122013843800742607353802703288649}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a^{2} + \frac{118211476248047691057153343795895937458499329123287123199942958192752998507581147918848}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131} a + \frac{164148978458600023587708817077624711200779101716025405513064042639751317029098886856477}{458021124744186876348154712294408718736845883692133859769654569503637448365188128771131}$

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:  \( 20835202759585965000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.361.1, 4.4.793117.1, 6.6.286315237.1, \(\Q(\zeta_{19})^+\), 12.12.65016888286672160858773.1, 18.18.3058776789325072365774692364013.1

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 $36$ $18^{2}$ $36$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ $36$ $18^{2}$ $36$ $18^{2}$ $36$

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
13Data not computed
19Data not computed