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Properties

Label 36.36.7824257734...0000.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 3^{48}\cdot 5^{27}\cdot 7^{24}$
Root discriminant $105.88$
Ramified primes $2, 3, 5, 7$
Class number Not computed
Class group Not computed
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![405901, -47460006, -129046575, 681310144, 1671476916, -4006725276, -9004822995, 12667223022, 26726967447, -24331055976, -49300418040, 30554706504, 60523405652, -26292514800, -51621469782, 15977803388, 31442289642, -6983909430, -13908705320, 2215904058, 4507244436, -510857842, -1071803142, 84929628, 186077966, -9998316, -23292630, 805868, 2056674, -41868, -123596, 1248, 4761, -16, -105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 105*x^34 - 16*x^33 + 4761*x^32 + 1248*x^31 - 123596*x^30 - 41868*x^29 + 2056674*x^28 + 805868*x^27 - 23292630*x^26 - 9998316*x^25 + 186077966*x^24 + 84929628*x^23 - 1071803142*x^22 - 510857842*x^21 + 4507244436*x^20 + 2215904058*x^19 - 13908705320*x^18 - 6983909430*x^17 + 31442289642*x^16 + 15977803388*x^15 - 51621469782*x^14 - 26292514800*x^13 + 60523405652*x^12 + 30554706504*x^11 - 49300418040*x^10 - 24331055976*x^9 + 26726967447*x^8 + 12667223022*x^7 - 9004822995*x^6 - 4006725276*x^5 + 1671476916*x^4 + 681310144*x^3 - 129046575*x^2 - 47460006*x + 405901)
 
gp: K = bnfinit(x^36 - 105*x^34 - 16*x^33 + 4761*x^32 + 1248*x^31 - 123596*x^30 - 41868*x^29 + 2056674*x^28 + 805868*x^27 - 23292630*x^26 - 9998316*x^25 + 186077966*x^24 + 84929628*x^23 - 1071803142*x^22 - 510857842*x^21 + 4507244436*x^20 + 2215904058*x^19 - 13908705320*x^18 - 6983909430*x^17 + 31442289642*x^16 + 15977803388*x^15 - 51621469782*x^14 - 26292514800*x^13 + 60523405652*x^12 + 30554706504*x^11 - 49300418040*x^10 - 24331055976*x^9 + 26726967447*x^8 + 12667223022*x^7 - 9004822995*x^6 - 4006725276*x^5 + 1671476916*x^4 + 681310144*x^3 - 129046575*x^2 - 47460006*x + 405901, 1)
 

Normalized defining polynomial

\( x^{36} - 105 x^{34} - 16 x^{33} + 4761 x^{32} + 1248 x^{31} - 123596 x^{30} - 41868 x^{29} + 2056674 x^{28} + 805868 x^{27} - 23292630 x^{26} - 9998316 x^{25} + 186077966 x^{24} + 84929628 x^{23} - 1071803142 x^{22} - 510857842 x^{21} + 4507244436 x^{20} + 2215904058 x^{19} - 13908705320 x^{18} - 6983909430 x^{17} + 31442289642 x^{16} + 15977803388 x^{15} - 51621469782 x^{14} - 26292514800 x^{13} + 60523405652 x^{12} + 30554706504 x^{11} - 49300418040 x^{10} - 24331055976 x^{9} + 26726967447 x^{8} + 12667223022 x^{7} - 9004822995 x^{6} - 4006725276 x^{5} + 1671476916 x^{4} + 681310144 x^{3} - 129046575 x^{2} - 47460006 x + 405901 \)

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:  \(7824257734407727258096221831729050732882821632000000000000000000000000000=2^{36}\cdot 3^{48}\cdot 5^{27}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.88$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(907,·)$, $\chi_{1260}(781,·)$, $\chi_{1260}(529,·)$, $\chi_{1260}(403,·)$, $\chi_{1260}(667,·)$, $\chi_{1260}(541,·)$, $\chi_{1260}(289,·)$, $\chi_{1260}(547,·)$, $\chi_{1260}(421,·)$, $\chi_{1260}(169,·)$, $\chi_{1260}(43,·)$, $\chi_{1260}(1201,·)$, $\chi_{1260}(949,·)$, $\chi_{1260}(823,·)$, $\chi_{1260}(1087,·)$, $\chi_{1260}(961,·)$, $\chi_{1260}(67,·)$, $\chi_{1260}(709,·)$, $\chi_{1260}(583,·)$, $\chi_{1260}(841,·)$, $\chi_{1260}(589,·)$, $\chi_{1260}(463,·)$, $\chi_{1260}(163,·)$, $\chi_{1260}(1129,·)$, $\chi_{1260}(1243,·)$, $\chi_{1260}(487,·)$, $\chi_{1260}(361,·)$, $\chi_{1260}(1003,·)$, $\chi_{1260}(109,·)$, $\chi_{1260}(1009,·)$, $\chi_{1260}(883,·)$, $\chi_{1260}(967,·)$, $\chi_{1260}(247,·)$, $\chi_{1260}(121,·)$, $\chi_{1260}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11}$, $\frac{1}{4} a^{28} - \frac{1}{4} a^{26} - \frac{1}{4} a^{22} - \frac{1}{4} a^{20} - \frac{1}{4} a^{16} + \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{29} - \frac{1}{4} a^{27} - \frac{1}{4} a^{23} - \frac{1}{4} a^{21} - \frac{1}{4} a^{17} + \frac{1}{4} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{30} - \frac{1}{4} a^{26} - \frac{1}{4} a^{24} - \frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{2} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{31} - \frac{1}{4} a^{27} - \frac{1}{4} a^{25} - \frac{1}{4} a^{21} - \frac{1}{4} a^{19} + \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{2032} a^{32} + \frac{29}{254} a^{31} - \frac{121}{1016} a^{30} - \frac{123}{1016} a^{29} - \frac{107}{1016} a^{28} + \frac{73}{1016} a^{27} + \frac{17}{254} a^{26} - \frac{11}{508} a^{25} + \frac{45}{508} a^{24} - \frac{53}{1016} a^{23} + \frac{215}{1016} a^{22} - \frac{15}{1016} a^{21} + \frac{33}{254} a^{20} + \frac{31}{254} a^{19} - \frac{31}{127} a^{18} + \frac{47}{508} a^{17} + \frac{51}{508} a^{16} + \frac{403}{1016} a^{15} - \frac{63}{508} a^{14} - \frac{62}{127} a^{13} + \frac{359}{1016} a^{12} + \frac{447}{1016} a^{11} + \frac{57}{127} a^{10} + \frac{235}{1016} a^{9} - \frac{259}{1016} a^{8} + \frac{233}{508} a^{7} + \frac{201}{1016} a^{6} - \frac{307}{1016} a^{5} - \frac{37}{2032} a^{4} - \frac{13}{508} a^{3} - \frac{247}{508} a^{2} - \frac{201}{508} a - \frac{643}{2032}$, $\frac{1}{2032} a^{33} - \frac{109}{1016} a^{31} + \frac{9}{1016} a^{30} - \frac{19}{1016} a^{29} + \frac{5}{1016} a^{28} - \frac{13}{127} a^{27} - \frac{25}{508} a^{26} + \frac{57}{508} a^{25} - \frac{105}{1016} a^{24} - \frac{189}{1016} a^{23} - \frac{111}{1016} a^{22} + \frac{7}{127} a^{21} - \frac{5}{254} a^{20} - \frac{15}{254} a^{19} + \frac{113}{508} a^{18} + \frac{69}{508} a^{17} + \frac{107}{1016} a^{16} + \frac{179}{508} a^{15} + \frac{36}{127} a^{14} + \frac{115}{1016} a^{13} + \frac{471}{1016} a^{12} - \frac{31}{254} a^{11} + \frac{107}{1016} a^{10} - \frac{423}{1016} a^{9} + \frac{51}{508} a^{8} - \frac{215}{1016} a^{7} - \frac{203}{1016} a^{6} + \frac{171}{2032} a^{5} - \frac{153}{508} a^{4} - \frac{25}{508} a^{3} + \frac{207}{508} a^{2} + \frac{973}{2032} a - \frac{11}{127}$, $\frac{1}{1565375159025646824801399717056563948647045008} a^{34} - \frac{19503383538328059568164832547549598180387}{782687579512823412400699858528281974323522504} a^{33} + \frac{211441385790990888795455475205409146838487}{1565375159025646824801399717056563948647045008} a^{32} + \frac{44884680509579956082290055886341443509606413}{782687579512823412400699858528281974323522504} a^{31} - \frac{8523393066200718394109023692369605067398785}{97835947439102926550087482316035246790440313} a^{30} + \frac{2926257898262013070758235707328292242838060}{97835947439102926550087482316035246790440313} a^{29} + \frac{43277697889682772569714942420534713836474149}{782687579512823412400699858528281974323522504} a^{28} + \frac{31792133727128102704539685310062076771882637}{782687579512823412400699858528281974323522504} a^{27} + \frac{79739790854537439719522252312397591171261057}{391343789756411706200349929264140987161761252} a^{26} + \frac{100682649707634194536877813017225572037377087}{782687579512823412400699858528281974323522504} a^{25} - \frac{110427511708986194028990411320660945862366295}{782687579512823412400699858528281974323522504} a^{24} + \frac{36758008096992289526789478372142709306083229}{391343789756411706200349929264140987161761252} a^{23} + \frac{78649951417340303542592722575719507639197935}{782687579512823412400699858528281974323522504} a^{22} + \frac{106186710545590101239953380632241802810862131}{782687579512823412400699858528281974323522504} a^{21} + \frac{31279179957385175484125576944053935579919845}{195671894878205853100174964632070493580880626} a^{20} + \frac{10964673081222967053026213438016534961639408}{97835947439102926550087482316035246790440313} a^{19} - \frac{23693994579969815158256647655884905055740218}{97835947439102926550087482316035246790440313} a^{18} + \frac{78506166050047647343867748371489943213627909}{782687579512823412400699858528281974323522504} a^{17} + \frac{19981294808350598492425736339150223412912393}{97835947439102926550087482316035246790440313} a^{16} - \frac{297823817945962310445107357300737898486916191}{782687579512823412400699858528281974323522504} a^{15} + \frac{169134132954847665389914554160259702638514813}{782687579512823412400699858528281974323522504} a^{14} + \frac{116165134517242313853867823196385404055100323}{782687579512823412400699858528281974323522504} a^{13} - \frac{76333798656440448980633363632646664360486949}{782687579512823412400699858528281974323522504} a^{12} - \frac{22053995790884379885625916091109698616114379}{391343789756411706200349929264140987161761252} a^{11} - \frac{164793301611255808068633249070791286637686383}{782687579512823412400699858528281974323522504} a^{10} + \frac{259655385902055252595437486415150666261425641}{782687579512823412400699858528281974323522504} a^{9} - \frac{80090648418873181639778869575442765584794189}{391343789756411706200349929264140987161761252} a^{8} - \frac{116809153046845945469578559232773357954040469}{782687579512823412400699858528281974323522504} a^{7} + \frac{394443824801022740167023027649573248590147933}{1565375159025646824801399717056563948647045008} a^{6} - \frac{5004692755933547119241176078740266519653231}{391343789756411706200349929264140987161761252} a^{5} + \frac{522605935158367564537119574279995446793030627}{1565375159025646824801399717056563948647045008} a^{4} - \frac{94638442872599947569670185973927021202356461}{195671894878205853100174964632070493580880626} a^{3} - \frac{274813721015557851205358660526053694731268107}{1565375159025646824801399717056563948647045008} a^{2} + \frac{283412027042599017011212639310799119108746101}{782687579512823412400699858528281974323522504} a - \frac{443615848150710368852516148150879633434669235}{1565375159025646824801399717056563948647045008}$, $\frac{1}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{35} + \frac{498693829797065419974429076410931325469863858088997}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{34} - \frac{360476819830536602340723119250946924835724431671165633061078248970506520199960866644977878467}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{33} + \frac{188470137970177255367672905570209417262813941049295472804913764935157017019395090039045702809}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{32} + \frac{18239607178689237271384928697728473782877778844550247614226755390965963881849694839898903426609}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{31} + \frac{63437372935941435630754024265430027268575309832211483857250600742507976347408700984352799466899}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{30} + \frac{589726882740015583573660712131439719726157202595880354468446156828885547981022863881466174427}{218207866132204105738300881308161605382793934103741146196841604492741771798032852826394768763386} a^{29} - \frac{66543741068889798490861877986038839137912591180467530265475634223739097830512286579719260503585}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{28} + \frac{57298951614145337133731275118876576906008799072122321592260418223605182258917066022976885263347}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{27} - \frac{119335104706424413863329510722353924350330955038083272810460279503503701242829389832723813335925}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{26} + \frac{39693463864639897768988715605091635660133183843116333961232128710424456697166694177008524480233}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{25} - \frac{215205675376524597291353432997927619730657103684471402101889499766658072715355530379448307574039}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{24} - \frac{83026345620014242146859097020365529024865554033478854551944088059116897775048841834611037266393}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{23} + \frac{206237809285668450447554593620524077230762079161870308273760965809951974448078074326546013999673}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{22} + \frac{101359793857188945602434865604548191800547145820801769820709184898114027279372352192301443656047}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{21} + \frac{25021699481241969073966172582107030173360453050712801786938115985646125141412993107335601893343}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{20} - \frac{95593002810752402184363459712573264993100196311435012874486647558142480802701014108137868158249}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{19} - \frac{175847687643045658410748236028562445885936294250959303311571560426673275787572186402881236846745}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{18} + \frac{10612564291810669063929199774525451347105172792471387863502274405142543402031163516144481176997}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{17} + \frac{90360230422150329671960388181259509818403964832968392259025042038588067571995009984376979250959}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{16} - \frac{358096862341815626543264630852099767933698779359999240741172623894129059567122869102171044387165}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{15} + \frac{119770451809266479222848311342615185294666745209650259479819100289136791651937508145404133644709}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{14} + \frac{114778437498817577569773284670361609159160534512954247090942047302280524449667048886359552554171}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{13} - \frac{42282658575023341775015681287322551451452083046216402679257991979193210614642593760497955359647}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{12} - \frac{20333840486685427695257048252339999188775973156397794260899154975934670878676454079860073417680}{109103933066102052869150440654080802691396967051870573098420802246370885899016426413197384381693} a^{11} + \frac{34018485307590101583722077520297515576035113153936630353494460053453217592491904122241958350562}{109103933066102052869150440654080802691396967051870573098420802246370885899016426413197384381693} a^{10} + \frac{56568839737087571379424715864567718657333785957852613429849360280421622076715369234750654796877}{218207866132204105738300881308161605382793934103741146196841604492741771798032852826394768763386} a^{9} - \frac{149525136773928021388181846690472425509289969060116755014367024136749489188765440699311153306841}{436415732264408211476601762616323210765587868207482292393683208985483543596065705652789537526772} a^{8} + \frac{808901606182928677418654564073793303209970909908030412851285962264463120772561808779991588354359}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{7} + \frac{37845744483131932531865327331559630914045020037089415713842286322531763043766592631552278457621}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{6} + \frac{491588375347394269524282220470068102753631216489270873608281273670426503154793241995335121563021}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{5} - \frac{186337284030628230952404014762840699905139271106084481503625530968186944661145373777878104436603}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544} a^{4} - \frac{780611677581245692737046482856169325178220919074790852736931964609297279870068892546330216173431}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{3} - \frac{387679042728100831365029172186292777653754062043726707595992895452718180893055856047668053567555}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a^{2} - \frac{424920014834695942030219150224665351717273278140237572164186445300172446385813889363591717450181}{1745662929057632845906407050465292843062351472829929169574732835941934174384262822611158150107088} a - \frac{386725897096075150026230780578667693600978765238589363663959321387137244174311686287322248900519}{872831464528816422953203525232646421531175736414964584787366417970967087192131411305579075053544}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{20})^+\), 6.6.820125.1, 6.6.300125.1, 6.6.1969120125.2, 6.6.1969120125.1, 9.9.62523502209.1, 12.12.344373768000000000.1, 12.12.46118408000000000.1, 12.12.1985246242140168000000000.2, 12.12.1985246242140168000000000.1, 18.18.7635133454060210702501953125.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
3Data not computed
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7Data not computed