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Properties

Label 36.36.1326595912...8853.1
Degree $36$
Signature $[36, 0]$
Discriminant $13^{33}\cdot 19^{30}$
Root discriminant $122.11$
Ramified primes $13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18773, 471498, -1884048, -35766984, 83002961, 475096085, -717542006, -2560846304, 3020402597, 7354733074, -7568837976, -12756349726, 12203357336, 14360136230, -13230545958, -10975518510, 9939296148, 5859024782, -5280367442, -2221244124, 2008422826, 602644282, -549690124, -116987282, 108034873, 16117232, -15099705, -1547896, 1472129, 100351, -96839, -4149, 4058, 98, -97, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 97*x^34 + 98*x^33 + 4058*x^32 - 4149*x^31 - 96839*x^30 + 100351*x^29 + 1472129*x^28 - 1547896*x^27 - 15099705*x^26 + 16117232*x^25 + 108034873*x^24 - 116987282*x^23 - 549690124*x^22 + 602644282*x^21 + 2008422826*x^20 - 2221244124*x^19 - 5280367442*x^18 + 5859024782*x^17 + 9939296148*x^16 - 10975518510*x^15 - 13230545958*x^14 + 14360136230*x^13 + 12203357336*x^12 - 12756349726*x^11 - 7568837976*x^10 + 7354733074*x^9 + 3020402597*x^8 - 2560846304*x^7 - 717542006*x^6 + 475096085*x^5 + 83002961*x^4 - 35766984*x^3 - 1884048*x^2 + 471498*x + 18773)
 
gp: K = bnfinit(x^36 - x^35 - 97*x^34 + 98*x^33 + 4058*x^32 - 4149*x^31 - 96839*x^30 + 100351*x^29 + 1472129*x^28 - 1547896*x^27 - 15099705*x^26 + 16117232*x^25 + 108034873*x^24 - 116987282*x^23 - 549690124*x^22 + 602644282*x^21 + 2008422826*x^20 - 2221244124*x^19 - 5280367442*x^18 + 5859024782*x^17 + 9939296148*x^16 - 10975518510*x^15 - 13230545958*x^14 + 14360136230*x^13 + 12203357336*x^12 - 12756349726*x^11 - 7568837976*x^10 + 7354733074*x^9 + 3020402597*x^8 - 2560846304*x^7 - 717542006*x^6 + 475096085*x^5 + 83002961*x^4 - 35766984*x^3 - 1884048*x^2 + 471498*x + 18773, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 97 x^{34} + 98 x^{33} + 4058 x^{32} - 4149 x^{31} - 96839 x^{30} + 100351 x^{29} + 1472129 x^{28} - 1547896 x^{27} - 15099705 x^{26} + 16117232 x^{25} + 108034873 x^{24} - 116987282 x^{23} - 549690124 x^{22} + 602644282 x^{21} + 2008422826 x^{20} - 2221244124 x^{19} - 5280367442 x^{18} + 5859024782 x^{17} + 9939296148 x^{16} - 10975518510 x^{15} - 13230545958 x^{14} + 14360136230 x^{13} + 12203357336 x^{12} - 12756349726 x^{11} - 7568837976 x^{10} + 7354733074 x^{9} + 3020402597 x^{8} - 2560846304 x^{7} - 717542006 x^{6} + 475096085 x^{5} + 83002961 x^{4} - 35766984 x^{3} - 1884048 x^{2} + 471498 x + 18773 \)

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:  \(1326595912169561221084879732571729733056225211740623097203928593533001998853=13^{33}\cdot 19^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.11$
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$$\chi_{247}(1,·)$, $\chi_{247}(132,·)$, $\chi_{247}(134,·)$, $\chi_{247}(8,·)$, $\chi_{247}(140,·)$, $\chi_{247}(141,·)$, $\chi_{247}(144,·)$, $\chi_{247}(145,·)$, $\chi_{247}(18,·)$, $\chi_{247}(151,·)$, $\chi_{247}(153,·)$, $\chi_{247}(30,·)$, $\chi_{247}(159,·)$, $\chi_{247}(164,·)$, $\chi_{247}(37,·)$, $\chi_{247}(172,·)$, $\chi_{247}(178,·)$, $\chi_{247}(46,·)$, $\chi_{247}(49,·)$, $\chi_{247}(50,·)$, $\chi_{247}(31,·)$, $\chi_{247}(189,·)$, $\chi_{247}(191,·)$, $\chi_{247}(64,·)$, $\chi_{247}(68,·)$, $\chi_{247}(202,·)$, $\chi_{247}(77,·)$, $\chi_{247}(84,·)$, $\chi_{247}(87,·)$, $\chi_{247}(220,·)$, $\chi_{247}(227,·)$, $\chi_{247}(235,·)$, $\chi_{247}(236,·)$, $\chi_{247}(240,·)$, $\chi_{247}(121,·)$, $\chi_{247}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{103} a^{29} - \frac{30}{103} a^{28} - \frac{31}{103} a^{27} - \frac{40}{103} a^{26} - \frac{15}{103} a^{25} + \frac{34}{103} a^{24} + \frac{9}{103} a^{23} - \frac{7}{103} a^{22} - \frac{40}{103} a^{20} - \frac{1}{103} a^{19} - \frac{28}{103} a^{18} - \frac{40}{103} a^{17} - \frac{44}{103} a^{16} - \frac{45}{103} a^{15} + \frac{16}{103} a^{14} - \frac{19}{103} a^{13} + \frac{6}{103} a^{12} + \frac{43}{103} a^{11} - \frac{30}{103} a^{10} - \frac{10}{103} a^{9} + \frac{9}{103} a^{8} - \frac{2}{103} a^{7} - \frac{21}{103} a^{6} + \frac{24}{103} a^{5} - \frac{43}{103} a^{4} + \frac{6}{103} a^{3} - \frac{22}{103} a^{2} - \frac{3}{103} a + \frac{21}{103}$, $\frac{1}{26129143} a^{30} - \frac{124947}{26129143} a^{29} + \frac{9010985}{26129143} a^{28} + \frac{7189399}{26129143} a^{27} + \frac{1372301}{26129143} a^{26} + \frac{10418875}{26129143} a^{25} + \frac{2876517}{26129143} a^{24} + \frac{6933121}{26129143} a^{23} - \frac{3097570}{26129143} a^{22} + \frac{508883}{26129143} a^{21} + \frac{4259096}{26129143} a^{20} + \frac{8778537}{26129143} a^{19} - \frac{2242863}{26129143} a^{18} - \frac{2567581}{26129143} a^{17} - \frac{12813801}{26129143} a^{16} + \frac{12686257}{26129143} a^{15} + \frac{11691245}{26129143} a^{14} - \frac{56951}{253681} a^{13} + \frac{6074085}{26129143} a^{12} + \frac{12894971}{26129143} a^{11} - \frac{9650431}{26129143} a^{10} - \frac{6591902}{26129143} a^{9} + \frac{123384}{26129143} a^{8} - \frac{9005664}{26129143} a^{7} + \frac{3551105}{26129143} a^{6} - \frac{1383011}{26129143} a^{5} - \frac{2715814}{26129143} a^{4} - \frac{206405}{26129143} a^{3} + \frac{12787581}{26129143} a^{2} - \frac{4699523}{26129143} a + \frac{3822998}{26129143}$, $\frac{1}{26129143} a^{31} - \frac{91919}{26129143} a^{29} - \frac{4069124}{26129143} a^{28} - \frac{4179025}{26129143} a^{27} + \frac{4459911}{26129143} a^{26} + \frac{2873872}{26129143} a^{25} + \frac{2847196}{26129143} a^{24} - \frac{10200342}{26129143} a^{23} - \frac{5435386}{26129143} a^{22} - \frac{10670765}{26129143} a^{21} + \frac{12504466}{26129143} a^{20} - \frac{3096712}{26129143} a^{19} - \frac{1438547}{26129143} a^{18} + \frac{3974244}{26129143} a^{17} + \frac{4308254}{26129143} a^{16} - \frac{6059643}{26129143} a^{15} + \frac{8820762}{26129143} a^{14} + \frac{6214027}{26129143} a^{13} - \frac{314051}{26129143} a^{12} - \frac{983456}{26129143} a^{11} + \frac{6667657}{26129143} a^{10} - \frac{4796361}{26129143} a^{9} - \frac{2904723}{26129143} a^{8} - \frac{8837619}{26129143} a^{7} + \frac{3361356}{26129143} a^{6} - \frac{6919185}{26129143} a^{5} + \frac{10220774}{26129143} a^{4} + \frac{7746248}{26129143} a^{3} - \frac{5527942}{26129143} a^{2} + \frac{2098754}{26129143} a + \frac{1462708}{26129143}$, $\frac{1}{26129143} a^{32} - \frac{113608}{26129143} a^{29} + \frac{9608276}{26129143} a^{28} - \frac{2833010}{26129143} a^{27} - \frac{7078189}{26129143} a^{26} - \frac{2614965}{26129143} a^{25} - \frac{10520580}{26129143} a^{24} - \frac{2073527}{26129143} a^{23} - \frac{4145833}{26129143} a^{22} - \frac{8774170}{26129143} a^{21} - \frac{3186333}{26129143} a^{20} + \frac{573941}{26129143} a^{19} + \frac{12350381}{26129143} a^{18} - \frac{5735385}{26129143} a^{17} - \frac{8112726}{26129143} a^{16} - \frac{12568009}{26129143} a^{15} - \frac{6939326}{26129143} a^{14} + \frac{790341}{26129143} a^{13} + \frac{381655}{26129143} a^{12} + \frac{4715183}{26129143} a^{11} - \frac{3726700}{26129143} a^{10} + \frac{11135190}{26129143} a^{9} + \frac{58645}{26129143} a^{8} - \frac{1289398}{26129143} a^{7} + \frac{10218427}{26129143} a^{6} - \frac{2321303}{26129143} a^{5} + \frac{9149318}{26129143} a^{4} - \frac{3237699}{26129143} a^{3} - \frac{5634721}{26129143} a^{2} - \frac{9536663}{26129143} a - \frac{13062518}{26129143}$, $\frac{1}{26129143} a^{33} - \frac{36342}{26129143} a^{29} + \frac{8784574}{26129143} a^{28} + \frac{3400158}{26129143} a^{27} - \frac{3736170}{26129143} a^{26} - \frac{3548401}{26129143} a^{25} + \frac{5718186}{26129143} a^{24} - \frac{8136276}{26129143} a^{23} + \frac{9324631}{26129143} a^{22} + \frac{12329215}{26129143} a^{21} - \frac{10543840}{26129143} a^{20} + \frac{3913201}{26129143} a^{19} - \frac{1766234}{26129143} a^{18} + \frac{7000546}{26129143} a^{17} - \frac{11663026}{26129143} a^{16} - \frac{9126427}{26129143} a^{15} + \frac{4636612}{26129143} a^{14} + \frac{5746489}{26129143} a^{13} + \frac{8083430}{26129143} a^{12} - \frac{3867311}{26129143} a^{11} + \frac{3008580}{26129143} a^{10} - \frac{3600481}{26129143} a^{9} + \frac{11914150}{26129143} a^{8} + \frac{13047005}{26129143} a^{7} + \frac{3989642}{26129143} a^{6} - \frac{12141009}{26129143} a^{5} + \frac{6960474}{26129143} a^{4} - \frac{7666493}{26129143} a^{3} - \frac{9252699}{26129143} a^{2} - \frac{11321110}{26129143} a - \frac{1800187}{26129143}$, $\frac{1}{26129143} a^{34} - \frac{28235}{26129143} a^{29} - \frac{6333112}{26129143} a^{28} + \frac{12160327}{26129143} a^{27} - \frac{8367912}{26129143} a^{26} - \frac{7202809}{26129143} a^{25} - \frac{7636666}{26129143} a^{24} + \frac{9989426}{26129143} a^{23} + \frac{1490466}{26129143} a^{22} + \frac{9978045}{26129143} a^{21} + \frac{2488435}{26129143} a^{20} + \frac{5449207}{26129143} a^{19} + \frac{6608148}{26129143} a^{18} - \frac{11970541}{26129143} a^{17} - \frac{5563307}{26129143} a^{16} - \frac{2676539}{26129143} a^{15} + \frac{6783171}{26129143} a^{14} + \frac{2613191}{26129143} a^{13} - \frac{6841778}{26129143} a^{12} + \frac{11192258}{26129143} a^{11} + \frac{1521685}{26129143} a^{10} - \frac{11181998}{26129143} a^{9} + \frac{3363099}{26129143} a^{8} - \frac{9544880}{26129143} a^{7} + \frac{6515208}{26129143} a^{6} + \frac{10689095}{26129143} a^{5} + \frac{4796072}{26129143} a^{4} + \frac{6398502}{26129143} a^{3} + \frac{12705595}{26129143} a^{2} + \frac{5463903}{26129143} a - \frac{9495599}{26129143}$, $\frac{1}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{35} - \frac{48961854949208561434524736119205094127551539308415118442907269336375937037684845268978662625425}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{34} - \frac{52671387921088118590600319775588747658393018750799438764886201331408574341993292003629242359441}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{33} - \frac{51152477689657533243625869167692038208483043762560128223664522733389501599020366036386038389774}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{32} - \frac{27801163742836379418669190478529781010467006338944118136930211541538974671277107673889402032617}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{31} + \frac{11894569091751821827951896402425521365758419954574005821123275208470309293486545204387238218630}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{30} - \frac{12691020640367968508410569785740873400706285525066323106522354162487286623066049435708485642029086519}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{29} - \frac{1301804207143443568940648417529363004437194006273770186573007928368068710892508448021555911733556298071}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{28} - \frac{250938425105930997794459476967624649980606026761096096138293957737368428004767126703398546981800819148}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{27} + \frac{380707716275871125060152456232482167694005677972821332266579984677435316740399052806553112688077316009}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{26} - \frac{1722338959211661329927556496352646698387536806916354424962001311772341301188539490028166080640271999924}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{25} - \frac{1589348352059464485881928139727005749549229117237146411558144291572887671886615884010242277920520480864}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{24} + \frac{841903866659949568228805743669289427392480074061951405954880378672979138213265784952780532578698057184}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{23} + \frac{898675530145672993797674274141969931584805050375832093348115302232922427806795761470512436728982765658}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{22} + \frac{359116302280712780678956012772921808303587636459290398103964397971474511485565334700754200955382188946}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{21} - \frac{696220612701856427833252663045739437017911324038534074204605373758541965576767228166724987313562611122}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{20} + \frac{531422787819915633586937608825510456966533777685686873543662596654492440561404351066668327576460857989}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{19} - \frac{356347481971186554455973637822492779881993512968610160239828389542154142282897331753064508517094447287}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{18} - \frac{915425797409005829397927898342775364147758540496891870389751629238849440401406345451252288398313813900}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{17} - \frac{1620466350028233889308317959075635184057956064388847739757206507051338678385944028034705066452348114163}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{16} + \frac{348822375885810808854324416508025036648045593879191794720775450179484953512964058533323145168394726797}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{15} - \frac{1417441382415742205601507370918447727757362940115902959059658238494385924400553610700302397547462707077}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{14} + \frac{687442382747327323561584538874442718719809270071381192100548579046902531585926572882223488764158167164}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{13} - \frac{329009085125142119894314931635883548116337578495093407453192870797384585427096396808979953419970910624}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{12} + \frac{159531382853783008122045485164599927627515185875311747140728251678623367813288337677091739708327004996}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{11} + \frac{470500133030873130388850177733828202628233829207702140418387427834368677859905457095904095166437277144}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{10} + \frac{990280827308512083083984424194891331080307162866091598097611524286762917926680634090795921110470046047}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{9} + \frac{1647284705138417191236529794933883989465227755752742311134599686510724398286265092060669399664437019146}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{8} - \frac{1721907364448867013119099855416442724964970799163382848192510753674160759840750267502917005796779623207}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{7} - \frac{519753617522152276929856790361199473970012030456970898457690810704295413606555787747604656361641864296}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{6} - \frac{343115484146810754667249756052725210661149465859044146907195288470979974820425756740116608609703470117}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{5} + \frac{848679878333859284170620084515756757220655007154126636988410383117466626906735819622841243684402089059}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{4} + \frac{179204235587270231981639589972317044984994887952733458558436502057635636956833595726270652120957617839}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{3} - \frac{339981732632365052306215680032167634186269764988378290614112731517336374793713395316386998268543431929}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a^{2} + \frac{904489593847788304939547168314711598224441948545217599281518866673793507909310506254755223046682246896}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437} a + \frac{1112659595986133632924982126057845495298765903696353274677374276510369509832720274744821141180759747524}{3465758642168072959349983088026054128203138566499295107977334861194421023562797306542357096177648849437}$

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:  \( 129475995156478280000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.361.1, 3.3.61009.2, 3.3.61009.1, 4.4.793117.1, \(\Q(\zeta_{13})^+\), 6.6.286315237.1, 6.6.48387275053.2, 6.6.48387275053.1, 9.9.227081481823729.1, 12.12.84313764630777811597.1, 12.12.65016888286672160858773.1, 12.12.10987854120447595185132637.2, 12.12.10987854120447595185132637.1, 18.18.113290500653811459555808941573877.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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