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Properties

Label 36.0.32387780912...9776.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 3^{48}\cdot 7^{30}$
Root discriminant $61.94$
Ramified primes $2, 3, 7$
Class number $31752$ (GRH)
Class group $[2, 126, 126]$ (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, 6, 57, 470, 4017, 34014, 288698, -713472, 1801707, -4002972, 8930712, -18279048, 48532934, -27478788, 39407814, -28196192, 29964870, -22048776, 16216396, 5079336, 6752154, 2565056, 2712360, 643836, 1143236, 103356, 119598, 13560, 12297, 1302, 1229, -54, 123, -2, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 12*x^34 - 2*x^33 + 123*x^32 - 54*x^31 + 1229*x^30 + 1302*x^29 + 12297*x^28 + 13560*x^27 + 119598*x^26 + 103356*x^25 + 1143236*x^24 + 643836*x^23 + 2712360*x^22 + 2565056*x^21 + 6752154*x^20 + 5079336*x^19 + 16216396*x^18 - 22048776*x^17 + 29964870*x^16 - 28196192*x^15 + 39407814*x^14 - 27478788*x^13 + 48532934*x^12 - 18279048*x^11 + 8930712*x^10 - 4002972*x^9 + 1801707*x^8 - 713472*x^7 + 288698*x^6 + 34014*x^5 + 4017*x^4 + 470*x^3 + 57*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^36 + 12*x^34 - 2*x^33 + 123*x^32 - 54*x^31 + 1229*x^30 + 1302*x^29 + 12297*x^28 + 13560*x^27 + 119598*x^26 + 103356*x^25 + 1143236*x^24 + 643836*x^23 + 2712360*x^22 + 2565056*x^21 + 6752154*x^20 + 5079336*x^19 + 16216396*x^18 - 22048776*x^17 + 29964870*x^16 - 28196192*x^15 + 39407814*x^14 - 27478788*x^13 + 48532934*x^12 - 18279048*x^11 + 8930712*x^10 - 4002972*x^9 + 1801707*x^8 - 713472*x^7 + 288698*x^6 + 34014*x^5 + 4017*x^4 + 470*x^3 + 57*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 12 x^{34} - 2 x^{33} + 123 x^{32} - 54 x^{31} + 1229 x^{30} + 1302 x^{29} + 12297 x^{28} + 13560 x^{27} + 119598 x^{26} + 103356 x^{25} + 1143236 x^{24} + 643836 x^{23} + 2712360 x^{22} + 2565056 x^{21} + 6752154 x^{20} + 5079336 x^{19} + 16216396 x^{18} - 22048776 x^{17} + 29964870 x^{16} - 28196192 x^{15} + 39407814 x^{14} - 27478788 x^{13} + 48532934 x^{12} - 18279048 x^{11} + 8930712 x^{10} - 4002972 x^{9} + 1801707 x^{8} - 713472 x^{7} + 288698 x^{6} + 34014 x^{5} + 4017 x^{4} + 470 x^{3} + 57 x^{2} + 6 x + 1 \)

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:  \(32387780912664470931540162651878867184371649203720968942991179776=2^{54}\cdot 3^{48}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.94$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(265,·)$, $\chi_{504}(13,·)$, $\chi_{504}(109,·)$, $\chi_{504}(145,·)$, $\chi_{504}(277,·)$, $\chi_{504}(409,·)$, $\chi_{504}(25,·)$, $\chi_{504}(157,·)$, $\chi_{504}(289,·)$, $\chi_{504}(37,·)$, $\chi_{504}(169,·)$, $\chi_{504}(433,·)$, $\chi_{504}(181,·)$, $\chi_{504}(73,·)$, $\chi_{504}(313,·)$, $\chi_{504}(61,·)$, $\chi_{504}(193,·)$, $\chi_{504}(325,·)$, $\chi_{504}(97,·)$, $\chi_{504}(457,·)$, $\chi_{504}(205,·)$, $\chi_{504}(397,·)$, $\chi_{504}(337,·)$, $\chi_{504}(85,·)$, $\chi_{504}(349,·)$, $\chi_{504}(421,·)$, $\chi_{504}(481,·)$, $\chi_{504}(229,·)$, $\chi_{504}(361,·)$, $\chi_{504}(493,·)$, $\chi_{504}(445,·)$, $\chi_{504}(241,·)$, $\chi_{504}(373,·)$, $\chi_{504}(121,·)$, $\chi_{504}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{13}$, $\frac{1}{4} a^{28} - \frac{1}{4}$, $\frac{1}{4} a^{29} - \frac{1}{4} a$, $\frac{1}{292} a^{30} - \frac{7}{292} a^{29} + \frac{3}{73} a^{28} + \frac{15}{73} a^{27} + \frac{16}{73} a^{26} + \frac{17}{146} a^{25} - \frac{18}{73} a^{24} - \frac{11}{73} a^{23} - \frac{5}{73} a^{22} + \frac{19}{146} a^{21} + \frac{13}{73} a^{20} - \frac{3}{73} a^{19} - \frac{2}{73} a^{18} + \frac{13}{73} a^{17} - \frac{3}{73} a^{16} + \frac{4}{73} a^{15} + \frac{11}{73} a^{14} + \frac{9}{73} a^{13} + \frac{34}{73} a^{12} - \frac{7}{146} a^{11} - \frac{14}{73} a^{10} + \frac{3}{73} a^{9} - \frac{35}{73} a^{8} + \frac{31}{146} a^{7} - \frac{4}{73} a^{6} + \frac{18}{73} a^{5} - \frac{33}{73} a^{4} + \frac{2}{73} a^{3} + \frac{35}{292} a^{2} - \frac{113}{292} a + \frac{25}{73}$, $\frac{1}{1551385517638746738916153209028} a^{31} + \frac{533950852267429754177889535}{1551385517638746738916153209028} a^{30} + \frac{120904967134139165521123554063}{1551385517638746738916153209028} a^{29} + \frac{3203705113604578525067337209}{775692758819373369458076604514} a^{28} + \frac{143126282838037536279005981447}{775692758819373369458076604514} a^{27} + \frac{105856199985151106725335803453}{775692758819373369458076604514} a^{26} + \frac{714487629510836525131065951}{10625928203005114650110638418} a^{25} + \frac{30384603836874094512386781089}{387846379409686684729038302257} a^{24} + \frac{67892044958219591769584343764}{387846379409686684729038302257} a^{23} + \frac{87279151358362142390443957216}{387846379409686684729038302257} a^{22} - \frac{39724894731603836339227908087}{387846379409686684729038302257} a^{21} - \frac{23978834765590447619518296476}{387846379409686684729038302257} a^{20} + \frac{187718936590878652908598007399}{775692758819373369458076604514} a^{19} - \frac{91047795703806036685976311027}{387846379409686684729038302257} a^{18} - \frac{155625095646997854308597043641}{775692758819373369458076604514} a^{17} + \frac{157489810848994599840047234459}{775692758819373369458076604514} a^{16} + \frac{48817669080417478872448772369}{775692758819373369458076604514} a^{15} - \frac{191138880360595266659356602763}{775692758819373369458076604514} a^{14} - \frac{222751031435400321067355473647}{775692758819373369458076604514} a^{13} - \frac{292776394926249485070079434657}{775692758819373369458076604514} a^{12} - \frac{150159454426792297140435823555}{775692758819373369458076604514} a^{11} - \frac{76655861687073995423787896760}{387846379409686684729038302257} a^{10} - \frac{114038078457048122919460435504}{387846379409686684729038302257} a^{9} - \frac{27040999108459149347123922913}{387846379409686684729038302257} a^{8} - \frac{35056415797122123721927624824}{387846379409686684729038302257} a^{7} + \frac{72138721514802425178575021192}{387846379409686684729038302257} a^{6} + \frac{171963645400782221876130297525}{775692758819373369458076604514} a^{5} + \frac{171979183056296804324141090104}{387846379409686684729038302257} a^{4} + \frac{154445797916852071602097185525}{1551385517638746738916153209028} a^{3} + \frac{598356988178048977044771299851}{1551385517638746738916153209028} a^{2} + \frac{135855032099573056447254827067}{1551385517638746738916153209028} a + \frac{183534472868907214815235507894}{387846379409686684729038302257}$, $\frac{1}{1551385517638746738916153209028} a^{32} - \frac{612979832500121758898690946}{387846379409686684729038302257} a^{30} + \frac{37522109760367643556511609697}{1551385517638746738916153209028} a^{29} - \frac{29423031960005844427137165429}{1551385517638746738916153209028} a^{28} - \frac{80130900758740168177186940147}{387846379409686684729038302257} a^{27} + \frac{2804020574722873059464784295}{387846379409686684729038302257} a^{26} - \frac{29473608992910141683608605160}{387846379409686684729038302257} a^{25} + \frac{600037173868895669976578152}{387846379409686684729038302257} a^{24} + \frac{21413934251657310431105321780}{387846379409686684729038302257} a^{23} + \frac{183918132182370660555410452541}{775692758819373369458076604514} a^{22} + \frac{2065142496812133197039162985}{387846379409686684729038302257} a^{21} + \frac{64186680242786202374745438978}{387846379409686684729038302257} a^{20} - \frac{120266608854044139708125749575}{775692758819373369458076604514} a^{19} - \frac{29179991660371291146881449953}{775692758819373369458076604514} a^{18} - \frac{132438280808886629639119552069}{775692758819373369458076604514} a^{17} + \frac{33443426196551864447832759039}{775692758819373369458076604514} a^{16} + \frac{16326817541426764300901821818}{387846379409686684729038302257} a^{15} + \frac{94859308702798283044898262870}{387846379409686684729038302257} a^{14} + \frac{29101485709223983044172665545}{387846379409686684729038302257} a^{13} - \frac{26312197345654519608465434646}{387846379409686684729038302257} a^{12} + \frac{137703906730774414484286513514}{387846379409686684729038302257} a^{11} + \frac{73767451687612651824460346170}{387846379409686684729038302257} a^{10} - \frac{1877974501609850616877519447}{5312964101502557325055319209} a^{9} + \frac{279259853336624182994306330987}{775692758819373369458076604514} a^{8} + \frac{156708614567674169239780388799}{387846379409686684729038302257} a^{7} - \frac{64062346505696722068857657859}{387846379409686684729038302257} a^{6} + \frac{205781290493510314877835819295}{775692758819373369458076604514} a^{5} + \frac{386882203785604088243411300945}{1551385517638746738916153209028} a^{4} + \frac{11815353993112541644302842089}{775692758819373369458076604514} a^{3} + \frac{226967328121763498315878075691}{775692758819373369458076604514} a^{2} + \frac{470076924865208729200271470927}{1551385517638746738916153209028} a - \frac{20347380757109140710699102995}{1551385517638746738916153209028}$, $\frac{1}{1551385517638746738916153209028} a^{33} + \frac{1051312169768987875537471127}{775692758819373369458076604514} a^{30} - \frac{148278981610906814552389431369}{1551385517638746738916153209028} a^{29} + \frac{12615746037227854506449653521}{775692758819373369458076604514} a^{28} + \frac{77839434519237848757184225175}{775692758819373369458076604514} a^{27} - \frac{22013324912674422800530786471}{775692758819373369458076604514} a^{26} - \frac{30769155055328674559286683500}{387846379409686684729038302257} a^{25} - \frac{67127075797521747128234784961}{387846379409686684729038302257} a^{24} + \frac{169905481175794311539530992529}{775692758819373369458076604514} a^{23} - \frac{15144890575230272873041773545}{387846379409686684729038302257} a^{22} - \frac{68865820478159403252662321377}{775692758819373369458076604514} a^{21} + \frac{136187106631947770890476020211}{775692758819373369458076604514} a^{20} + \frac{513183128196588861739142633}{3053908499288871533299514191} a^{19} + \frac{90836147820494100178020942309}{775692758819373369458076604514} a^{18} - \frac{53050682336339302848163309989}{775692758819373369458076604514} a^{17} - \frac{22275429410122133408417560401}{387846379409686684729038302257} a^{16} + \frac{5954221095568721483603535049}{387846379409686684729038302257} a^{15} + \frac{60399504568605561794097223720}{387846379409686684729038302257} a^{14} + \frac{155706221816858809407124562963}{775692758819373369458076604514} a^{13} + \frac{255096653620226240035127575943}{775692758819373369458076604514} a^{12} - \frac{174089034474654981097293353922}{387846379409686684729038302257} a^{11} + \frac{14771643248617931441046589213}{387846379409686684729038302257} a^{10} - \frac{210030239801003135734626968623}{775692758819373369458076604514} a^{9} + \frac{99350651356089640980453700383}{387846379409686684729038302257} a^{8} + \frac{254645262147007742870663573645}{775692758819373369458076604514} a^{7} - \frac{125924389893949669069625804585}{775692758819373369458076604514} a^{6} + \frac{348390994584575829976241775019}{1551385517638746738916153209028} a^{5} + \frac{138617097652005143011460701109}{775692758819373369458076604514} a^{4} + \frac{272933486571521973077183010553}{775692758819373369458076604514} a^{3} - \frac{128913630253504430705246955693}{775692758819373369458076604514} a^{2} + \frac{666354659893569791713546994493}{1551385517638746738916153209028} a + \frac{30655110030104678808257493759}{775692758819373369458076604514}$, $\frac{1}{1551385517638746738916153209028} a^{34} - \frac{444665052272797441099885853}{387846379409686684729038302257} a^{30} - \frac{47357427030475964499973361115}{387846379409686684729038302257} a^{29} - \frac{21343922509094277172794520943}{1551385517638746738916153209028} a^{28} + \frac{116982225711248034637162919255}{775692758819373369458076604514} a^{27} + \frac{40021052631397843744660762314}{387846379409686684729038302257} a^{26} + \frac{458741896424320527623149391}{10625928203005114650110638418} a^{25} - \frac{50771562996842133881298599891}{775692758819373369458076604514} a^{24} - \frac{116646525395352073049778856503}{775692758819373369458076604514} a^{23} - \frac{41907445339321495285618510861}{387846379409686684729038302257} a^{22} - \frac{18829317688214979802786754814}{387846379409686684729038302257} a^{21} + \frac{7276924671875882091706450278}{387846379409686684729038302257} a^{20} + \frac{88171225052214046675499802909}{387846379409686684729038302257} a^{19} - \frac{95654906021210785415566915463}{775692758819373369458076604514} a^{18} - \frac{155248152862054069810356455069}{775692758819373369458076604514} a^{17} - \frac{45301384545082645817082089899}{775692758819373369458076604514} a^{16} + \frac{58915792713068723172365454044}{387846379409686684729038302257} a^{15} - \frac{43560892473647281838028282793}{775692758819373369458076604514} a^{14} - \frac{103088993446542435708814678383}{775692758819373369458076604514} a^{13} + \frac{126538223621216775700708944312}{387846379409686684729038302257} a^{12} + \frac{303191108681948735560422792703}{775692758819373369458076604514} a^{11} + \frac{314049398280022751574301947313}{775692758819373369458076604514} a^{10} + \frac{278235533479583199773478403647}{775692758819373369458076604514} a^{9} - \frac{168060742253019847248483276125}{387846379409686684729038302257} a^{8} + \frac{76007978854249644726736261923}{387846379409686684729038302257} a^{7} + \frac{543880459574335422883704320015}{1551385517638746738916153209028} a^{6} - \frac{136730234819561944972186175013}{387846379409686684729038302257} a^{5} - \frac{231657764843253786886843412883}{775692758819373369458076604514} a^{4} + \frac{379472104367527390595848585113}{775692758819373369458076604514} a^{3} + \frac{353872471506123539716848969415}{775692758819373369458076604514} a^{2} + \frac{191309095634467054442647446430}{387846379409686684729038302257} a + \frac{362071388033628936374585069557}{1551385517638746738916153209028}$, $\frac{1}{1551385517638746738916153209028} a^{35} + \frac{44121338002259969342504567916}{387846379409686684729038302257} a^{28} - \frac{158276490865864602958953240867}{775692758819373369458076604514} a^{21} - \frac{94152286124785884567550089342}{387846379409686684729038302257} a^{14} - \frac{327600440761600381626869842539}{1551385517638746738916153209028} a^{7} + \frac{126915032107453675421511634406}{387846379409686684729038302257}$

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

Class group and class number

$C_{2}\times C_{126}\times C_{126}$, which has order $31752$ (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{1357407863507143437572574}{5312964101502557325055319209} a^{35} - \frac{226234643917857239595429}{5312964101502557325055319209} a^{34} + \frac{27826861201896440470237767}{10625928203005114650110638418} a^{33} - \frac{6108335385782145469076583}{5312964101502557325055319209} a^{32} + \frac{278042377375046547462782241}{10625928203005114650110638418} a^{31} - \frac{209040870752876891049638197}{10625928203005114650110638418} a^{30} + \frac{2782007416257890475304990413}{10625928203005114650110638418} a^{29} + \frac{1533870885763072084457008620}{5312964101502557325055319209} a^{28} + \frac{13528605471643945070567058771}{5312964101502557325055319209} a^{27} + \frac{11691353928387026427812579862}{5312964101502557325055319209} a^{26} + \frac{129319794687037719583059934122}{5312964101502557325055319209} a^{25} + \frac{72829004100748766856081312822}{5312964101502557325055319209} a^{24} + \frac{1236541825343058769275954328212}{5312964101502557325055319209} a^{23} + \frac{290152265394681609783846364512}{5312964101502557325055319209} a^{22} + \frac{763785577934267715881617152033}{5312964101502557325055319209} a^{21} + \frac{574560885649576659968843977572}{5312964101502557325055319209} a^{20} + \frac{1834355287345482234373178226942}{5312964101502557325055319209} a^{19} - \frac{2494098493592298337908972322452}{5312964101502557325055319209} a^{18} + \frac{3389545847247441431517941289615}{5312964101502557325055319209} a^{17} - \frac{99911973345380791537171812253143}{10625928203005114650110638418} a^{16} + \frac{4457706383935574688265050641103}{5312964101502557325055319209} a^{15} - \frac{3108326909237144250553999630026}{5312964101502557325055319209} a^{14} + \frac{5489915520889433415353571179343}{5312964101502557325055319209} a^{13} - \frac{2067676957718710269856173635796}{5312964101502557325055319209} a^{12} + \frac{1010218224626467331970886457724}{5312964101502557325055319209} a^{11} - \frac{452805472516576415048896807494}{5312964101502557325055319209} a^{10} - \frac{75934887454333794388468405669217}{5312964101502557325055319209} a^{9} - \frac{80706041932680720224314959744}{5312964101502557325055319209} a^{8} + \frac{32656744614898774678360580721}{5312964101502557325055319209} a^{7} + \frac{3847572589110998073799461003}{5312964101502557325055319209} a^{6} + \frac{908784564618032531454838293}{10625928203005114650110638418} a^{5} + \frac{53165141320696451304925815}{5312964101502557325055319209} a^{4} + \frac{12895374703317862656939453}{10625928203005114650110638418} a^{3} - \frac{382351649358295135882102155561}{5312964101502557325055319209} a^{2} + \frac{226234643917857239595429}{10625928203005114650110638418} a \) (order $14$)
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:  \( 160258501280890.78 \) (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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\sqrt{2}, \sqrt{-7})\), 6.0.2250423.1, \(\Q(\zeta_{7})\), 6.0.110270727.2, 6.0.110270727.1, 6.0.1152216576.2, 6.6.3359232.1, 6.0.8605184.1, 6.6.1229312.1, 6.0.56458612224.1, 6.6.8065516032.1, 6.0.56458612224.2, 6.6.8065516032.2, 9.9.62523502209.1, 12.0.1327603038009163776.2, 12.0.74049191673856.2, 12.0.3187574894260002226176.4, 12.0.3187574894260002226176.1, 18.0.1340851596668237962730583.1, 18.0.179966054889983269121047526375424.4, 18.18.524682375772545974113841184768.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
7Data not computed