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Properties

Label 36.0.47398461013...7721.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{30}\cdot 13^{24}$
Root discriminant $48.47$
Ramified primes $3, 7, 13$
Class number $468$ (GRH)
Class group $[6, 78]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 1, -50, -196, -49, 2434, 891, -32211, -127114, -31870, 1579124, 6189592, 1547178, -2057459, -2036590, -496621, 713006, 632324, -452289, -438826, -88085, 138311, 157678, 41632, -37236, 8000, 1716, -2131, 606, 289, -64, -16, 15, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 4*x^34 + 15*x^33 - 16*x^32 - 64*x^31 + 289*x^30 + 606*x^29 - 2131*x^28 + 1716*x^27 + 8000*x^26 - 37236*x^25 + 41632*x^24 + 157678*x^23 + 138311*x^22 - 88085*x^21 - 438826*x^20 - 452289*x^19 + 632324*x^18 + 713006*x^17 - 496621*x^16 - 2036590*x^15 - 2057459*x^14 + 1547178*x^13 + 6189592*x^12 + 1579124*x^11 - 31870*x^10 - 127114*x^9 - 32211*x^8 + 891*x^7 + 2434*x^6 - 49*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 4*x^34 + 15*x^33 - 16*x^32 - 64*x^31 + 289*x^30 + 606*x^29 - 2131*x^28 + 1716*x^27 + 8000*x^26 - 37236*x^25 + 41632*x^24 + 157678*x^23 + 138311*x^22 - 88085*x^21 - 438826*x^20 - 452289*x^19 + 632324*x^18 + 713006*x^17 - 496621*x^16 - 2036590*x^15 - 2057459*x^14 + 1547178*x^13 + 6189592*x^12 + 1579124*x^11 - 31870*x^10 - 127114*x^9 - 32211*x^8 + 891*x^7 + 2434*x^6 - 49*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 4 x^{34} + 15 x^{33} - 16 x^{32} - 64 x^{31} + 289 x^{30} + 606 x^{29} - 2131 x^{28} + 1716 x^{27} + 8000 x^{26} - 37236 x^{25} + 41632 x^{24} + 157678 x^{23} + 138311 x^{22} - 88085 x^{21} - 438826 x^{20} - 452289 x^{19} + 632324 x^{18} + 713006 x^{17} - 496621 x^{16} - 2036590 x^{15} - 2057459 x^{14} + 1547178 x^{13} + 6189592 x^{12} + 1579124 x^{11} - 31870 x^{10} - 127114 x^{9} - 32211 x^{8} + 891 x^{7} + 2434 x^{6} - 49 x^{5} - 196 x^{4} - 50 x^{3} + x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4739846101393836610854424577149665214795350880765994711657721=3^{18}\cdot 7^{30}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.47$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(131,·)$, $\chi_{273}(263,·)$, $\chi_{273}(139,·)$, $\chi_{273}(269,·)$, $\chi_{273}(16,·)$, $\chi_{273}(146,·)$, $\chi_{273}(107,·)$, $\chi_{273}(22,·)$, $\chi_{273}(152,·)$, $\chi_{273}(29,·)$, $\chi_{273}(40,·)$, $\chi_{273}(170,·)$, $\chi_{273}(172,·)$, $\chi_{273}(157,·)$, $\chi_{273}(178,·)$, $\chi_{273}(53,·)$, $\chi_{273}(55,·)$, $\chi_{273}(185,·)$, $\chi_{273}(61,·)$, $\chi_{273}(191,·)$, $\chi_{273}(68,·)$, $\chi_{273}(74,·)$, $\chi_{273}(79,·)$, $\chi_{273}(209,·)$, $\chi_{273}(211,·)$, $\chi_{273}(92,·)$, $\chi_{273}(94,·)$, $\chi_{273}(100,·)$, $\chi_{273}(230,·)$, $\chi_{273}(235,·)$, $\chi_{273}(113,·)$, $\chi_{273}(118,·)$, $\chi_{273}(248,·)$, $\chi_{273}(250,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{21} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{4}$, $\frac{1}{154646} a^{26} - \frac{30525}{154646} a^{25} + \frac{465}{77323} a^{23} - \frac{10709}{154646} a^{22} + \frac{14347}{154646} a^{20} + \frac{4052}{77323} a^{19} - \frac{30995}{154646} a^{18} - \frac{34169}{154646} a^{17} + \frac{36389}{154646} a^{16} + \frac{16129}{154646} a^{15} + \frac{2583}{154646} a^{14} + \frac{51619}{154646} a^{13} - \frac{24867}{77323} a^{12} + \frac{38947}{154646} a^{11} - \frac{11913}{154646} a^{10} + \frac{63857}{154646} a^{9} - \frac{43777}{154646} a^{8} - \frac{21901}{154646} a^{7} - \frac{74377}{154646} a^{6} - \frac{1}{2} a^{5} + \frac{23399}{77323} a^{4} + \frac{33475}{154646} a^{3} - \frac{1}{2} a^{2} + \frac{33307}{77323} a - \frac{29419}{154646}$, $\frac{1}{154646} a^{27} - \frac{33475}{154646} a^{25} + \frac{465}{77323} a^{24} + \frac{29419}{154646} a^{22} + \frac{14347}{154646} a^{21} - \frac{7193}{154646} a^{20} - \frac{6336}{77323} a^{19} - \frac{16158}{77323} a^{18} + \frac{37611}{154646} a^{17} - \frac{15932}{77323} a^{16} + \frac{24767}{154646} a^{15} + \frac{14117}{77323} a^{14} - \frac{67853}{154646} a^{13} - \frac{4472}{77323} a^{12} - \frac{36593}{77323} a^{11} + \frac{69601}{154646} a^{10} + \frac{16482}{77323} a^{9} + \frac{58583}{154646} a^{8} - \frac{33872}{77323} a^{7} - \frac{38661}{77323} a^{6} + \frac{23399}{77323} a^{5} - \frac{1}{2} a^{4} - \frac{76393}{154646} a^{3} + \frac{33307}{77323} a^{2} + \frac{14347}{154646}$, $\frac{1}{309292} a^{28} + \frac{35065}{154646} a^{21} + \frac{28281}{309292} a^{14} + \frac{10178}{77323} a^{7} - \frac{15297}{309292}$, $\frac{1}{309292} a^{29} + \frac{35065}{154646} a^{22} + \frac{28281}{309292} a^{15} + \frac{10178}{77323} a^{8} - \frac{15297}{309292} a$, $\frac{1}{24434068} a^{30} - \frac{5}{24434068} a^{29} - \frac{12}{6108517} a^{27} - \frac{19}{6108517} a^{26} + \frac{681}{77323} a^{25} + \frac{2529339}{12217034} a^{24} - \frac{966675}{6108517} a^{23} + \frac{1502548}{6108517} a^{22} + \frac{1898039}{12217034} a^{21} - \frac{804861}{6108517} a^{20} - \frac{1239080}{6108517} a^{19} - \frac{1342993}{6108517} a^{18} - \frac{111413}{6108517} a^{17} + \frac{1111879}{24434068} a^{16} + \frac{691541}{24434068} a^{15} - \frac{3018137}{12217034} a^{14} - \frac{707786}{6108517} a^{13} - \frac{648832}{6108517} a^{12} + \frac{602177}{6108517} a^{11} + \frac{782949}{6108517} a^{10} - \frac{2037501}{12217034} a^{9} - \frac{1158737}{12217034} a^{8} + \frac{5009753}{12217034} a^{7} - \frac{2994260}{6108517} a^{6} - \frac{26684}{77323} a^{5} + \frac{1739820}{6108517} a^{4} - \frac{1990277}{12217034} a^{3} - \frac{130715}{309292} a^{2} + \frac{12024881}{24434068} a + \frac{2938456}{6108517}$, $\frac{1}{19401466938735156388} a^{31} - \frac{40013245719}{19401466938735156388} a^{30} + \frac{10190852673609}{9700733469367578194} a^{29} - \frac{20181639118617}{19401466938735156388} a^{28} + \frac{10872578899319}{4850366734683789097} a^{27} + \frac{13712757018957}{4850366734683789097} a^{26} - \frac{304803099802702565}{9700733469367578194} a^{25} + \frac{911784319916013444}{4850366734683789097} a^{24} - \frac{1117049634113625853}{4850366734683789097} a^{23} - \frac{478150395033638300}{4850366734683789097} a^{22} - \frac{1461856983968146631}{9700733469367578194} a^{21} - \frac{2232644071668977447}{9700733469367578194} a^{20} + \frac{1712597438682803863}{9700733469367578194} a^{19} - \frac{296588501234512487}{4850366734683789097} a^{18} + \frac{2745271621970307711}{19401466938735156388} a^{17} - \frac{4170258316374111055}{19401466938735156388} a^{16} - \frac{1022451904659333713}{9700733469367578194} a^{15} - \frac{2276664905058421567}{19401466938735156388} a^{14} - \frac{58796472665023367}{9700733469367578194} a^{13} + \frac{1249809077919117177}{9700733469367578194} a^{12} + \frac{920400789734224351}{4850366734683789097} a^{11} - \frac{1811418290330190839}{9700733469367578194} a^{10} + \frac{1460064624279340292}{4850366734683789097} a^{9} + \frac{125872179537357339}{4850366734683789097} a^{8} + \frac{228776095303389883}{9700733469367578194} a^{7} + \frac{3368090295163641245}{9700733469367578194} a^{6} + \frac{4611041018068663083}{9700733469367578194} a^{5} + \frac{1965746603516047133}{9700733469367578194} a^{4} + \frac{3720735142933052779}{19401466938735156388} a^{3} - \frac{1565260999630418635}{19401466938735156388} a^{2} + \frac{4382380873262835643}{9700733469367578194} a - \frac{7434296442034653253}{19401466938735156388}$, $\frac{1}{19401466938735156388} a^{32} - \frac{29844322116}{4850366734683789097} a^{30} + \frac{5375221811558}{4850366734683789097} a^{29} - \frac{264607605113}{245588189097913372} a^{28} - \frac{12294028906737}{9700733469367578194} a^{27} + \frac{2247369814826}{4850366734683789097} a^{26} + \frac{70322133496900911}{9700733469367578194} a^{25} + \frac{158190190188885815}{9700733469367578194} a^{24} + \frac{622834409321031205}{4850366734683789097} a^{23} - \frac{1870554873677571927}{9700733469367578194} a^{22} + \frac{193458463409137772}{4850366734683789097} a^{21} + \frac{998390572265769031}{4850366734683789097} a^{20} + \frac{788846352054210294}{4850366734683789097} a^{19} - \frac{1358017051204568685}{19401466938735156388} a^{18} + \frac{1367518704449448511}{9700733469367578194} a^{17} + \frac{992609445929796691}{4850366734683789097} a^{16} + \frac{1126640955557596626}{4850366734683789097} a^{15} + \frac{1648239885704724275}{19401466938735156388} a^{14} + \frac{76562252910672743}{4850366734683789097} a^{13} - \frac{1690094997782133316}{4850366734683789097} a^{12} + \frac{4040547693698018421}{9700733469367578194} a^{11} + \frac{1979695265764450489}{9700733469367578194} a^{10} - \frac{2365454390067340271}{4850366734683789097} a^{9} + \frac{499663258173417502}{4850366734683789097} a^{8} - \frac{945777037165104913}{9700733469367578194} a^{7} + \frac{2340848393875967379}{9700733469367578194} a^{6} - \frac{27455635576275945}{61397047274478343} a^{5} - \frac{2848659044163239523}{19401466938735156388} a^{4} + \frac{1676413507391361069}{4850366734683789097} a^{3} + \frac{27203495215096618}{61397047274478343} a^{2} - \frac{3022458214072098401}{9700733469367578194} a - \frac{3059149797399508569}{19401466938735156388}$, $\frac{1}{19401466938735156388} a^{33} + \frac{36872095407}{4850366734683789097} a^{30} + \frac{3580639514031}{4850366734683789097} a^{29} - \frac{47658227735}{61397047274478343} a^{28} + \frac{3530390658973}{9700733469367578194} a^{27} - \frac{4374888716209}{9700733469367578194} a^{26} - \frac{22524283439715985}{122794094548956686} a^{25} - \frac{27490661838422887}{122794094548956686} a^{24} - \frac{49195939960535594}{4850366734683789097} a^{23} - \frac{743872049494138107}{4850366734683789097} a^{22} + \frac{1089284429247734841}{9700733469367578194} a^{21} - \frac{535137461403458499}{9700733469367578194} a^{20} - \frac{3236842735815484045}{19401466938735156388} a^{19} + \frac{1541236680637937567}{9700733469367578194} a^{18} - \frac{1185163249269930896}{4850366734683789097} a^{17} + \frac{623909924635619403}{9700733469367578194} a^{16} - \frac{598611567381207235}{9700733469367578194} a^{15} - \frac{2146694654678531755}{9700733469367578194} a^{14} - \frac{245542120633522819}{9700733469367578194} a^{13} - \frac{1269012793078608141}{9700733469367578194} a^{12} - \frac{3286617837987714637}{9700733469367578194} a^{11} + \frac{2254058281714461658}{4850366734683789097} a^{10} - \frac{3897084425072555777}{9700733469367578194} a^{9} - \frac{2590160683824514229}{9700733469367578194} a^{8} + \frac{1436072748903649977}{9700733469367578194} a^{7} + \frac{1033444645839457543}{4850366734683789097} a^{6} - \frac{24763316642955073}{245588189097913372} a^{5} + \frac{1280798283372564411}{4850366734683789097} a^{4} + \frac{3858818437881291093}{9700733469367578194} a^{3} - \frac{7835005464090403}{122794094548956686} a^{2} - \frac{3323535230392558423}{9700733469367578194} a + \frac{2284649066144210802}{4850366734683789097}$, $\frac{1}{19401466938735156388} a^{34} + \frac{70311001399}{4850366734683789097} a^{30} + \frac{211277245113}{9700733469367578194} a^{29} - \frac{1828774518207}{19401466938735156388} a^{28} + \frac{14438685793087}{9700733469367578194} a^{27} + \frac{3028536673899}{9700733469367578194} a^{26} - \frac{1036699521376669439}{9700733469367578194} a^{25} + \frac{3710048942798964}{4850366734683789097} a^{24} - \frac{1690933366204489777}{9700733469367578194} a^{23} - \frac{1880749360415910319}{9700733469367578194} a^{22} + \frac{1678593601443945381}{9700733469367578194} a^{21} - \frac{3660393260339793363}{19401466938735156388} a^{20} - \frac{1119634388861495281}{9700733469367578194} a^{19} + \frac{1202415434973055124}{4850366734683789097} a^{18} - \frac{218314136281031939}{4850366734683789097} a^{17} + \frac{2373890450353670645}{9700733469367578194} a^{16} + \frac{531789492947564975}{4850366734683789097} a^{15} - \frac{1839141531963849077}{19401466938735156388} a^{14} - \frac{1067012148068709064}{4850366734683789097} a^{13} - \frac{2559739384402536077}{9700733469367578194} a^{12} + \frac{1423161661580231829}{4850366734683789097} a^{11} + \frac{1006791145569718489}{4850366734683789097} a^{10} - \frac{1329770521544539967}{9700733469367578194} a^{9} + \frac{407621943914285981}{9700733469367578194} a^{8} + \frac{1475508464551878351}{9700733469367578194} a^{7} + \frac{5141439639541581375}{19401466938735156388} a^{6} + \frac{2160197698287659018}{4850366734683789097} a^{5} + \frac{1256510512278768227}{9700733469367578194} a^{4} + \frac{1515975004212601064}{4850366734683789097} a^{3} + \frac{482888463443247629}{4850366734683789097} a^{2} - \frac{1190731098367219679}{9700733469367578194} a + \frac{1432034254934672029}{19401466938735156388}$, $\frac{1}{19401466938735156388} a^{35} + \frac{27473905485521}{19401466938735156388} a^{28} - \frac{4027571870065511}{19401466938735156388} a^{21} - \frac{330448767479753295}{19401466938735156388} a^{14} + \frac{2438924830211536249}{19401466938735156388} a^{7} - \frac{422527189271961119}{19401466938735156388}$

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

Class group and class number

$C_{6}\times C_{78}$, which has order $468$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1455421144187393712}{4850366734683789097} a^{35} + \frac{1819276430234242140}{4850366734683789097} a^{34} + \frac{21831317212578027623}{19401466938735156388} a^{33} - \frac{23286738306998299392}{4850366734683789097} a^{32} + \frac{363855286046848428}{61397047274478343} a^{31} + \frac{87325268651243622720}{4850366734683789097} a^{30} - \frac{443903448977155082160}{4850366734683789097} a^{29} - \frac{776831035710021393780}{4850366734683789097} a^{28} + \frac{3321998761607726147640}{4850366734683789097} a^{27} - \frac{6545756573077635575515}{9700733469367578194} a^{26} - \frac{11018993482642757793552}{4850366734683789097} a^{25} + \frac{57104904013336579684032}{4850366734683789097} a^{24} - \frac{74140608506050023082992}{4850366734683789097} a^{23} - \frac{214339871904477471966240}{4850366734683789097} a^{22} - \frac{143928780080407645270248}{4850366734683789097} a^{21} + \frac{178525959954172228046628}{4850366734683789097} a^{20} + \frac{2426505826754486005680045}{19401466938735156388} a^{19} + \frac{498601814128577806341240}{4850366734683789097} a^{18} - \frac{1084865463047992570198380}{4850366734683789097} a^{17} - \frac{807649578438189455631600}{4850366734683789097} a^{16} + \frac{982223706130606862901720}{4850366734683789097} a^{15} + \frac{2783397972028732246760292}{4850366734683789097} a^{14} + \frac{2253445294888499839317288}{4850366734683789097} a^{13} - \frac{6000825813949206291090991}{9700733469367578194} a^{12} - \frac{8445514176937747763509320}{4850366734683789097} a^{11} - \frac{46174691220489252906912}{4850366734683789097} a^{10} + \frac{620956886588695714618512}{4850366734683789097} a^{9} + \frac{173408335355923304906808}{4850366734683789097} a^{8} + \frac{629469644861047780440}{4850366734683789097} a^{7} - \frac{13016922858326002511700}{4850366734683789097} a^{6} - \frac{12873200020809984580977}{19401466938735156388} a^{5} + \frac{956939402303211365640}{4850366734683789097} a^{4} + \frac{267433635244433594580}{4850366734683789097} a^{3} + \frac{1455421144187393712}{4850366734683789097} a^{2} - \frac{19648185446529815112}{4850366734683789097} a - \frac{5457829290702726420}{4850366734683789097} \) (order $42$)
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:  \( 118794051645799.06 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

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 $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.771147.1, 6.0.1851523947.1, 6.0.64827.1, 6.0.1851523947.2, 6.0.9796423.1, 6.6.264503421.1, 6.0.480024727.2, 6.6.12960667629.2, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 6.0.480024727.1, 6.6.12960667629.1, 9.9.567869252041.1, 12.0.69962059720703241.1, 12.0.167978905389408481641.2, \(\Q(\zeta_{21})\), 12.0.167978905389408481641.1, 18.0.6347285018761982937208599123.3, 18.0.110609092182866440454328583.1, 18.18.2177118761435360147462549499189.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$