Properties

Label 43.43.9952594992...9529.1
Degree $43$
Signature $[43, 0]$
Discriminant $173^{42}$
Root discriminant $153.46$
Ramified prime $173$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{43}$ (as 43T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -54, -370, 12893, 90620, -1134285, -2599666, 27505221, 37041429, -294221159, -287529242, 1732796449, 1310830451, -6269987218, -3755010451, 14834334408, 7121607714, -23896748020, -9297003415, 27001558938, 8590544711, -21895326590, -5730420663, 12962901258, 2797394685, -5671315301, -1007873658, 1846875875, 268953093, -448758019, -53056499, 81126975, 7671648, -10810701, -798942, 1042773, 58076, -70521, -2786, 3160, 79, -84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^43 - x^42 - 84*x^41 + 79*x^40 + 3160*x^39 - 2786*x^38 - 70521*x^37 + 58076*x^36 + 1042773*x^35 - 798942*x^34 - 10810701*x^33 + 7671648*x^32 + 81126975*x^31 - 53056499*x^30 - 448758019*x^29 + 268953093*x^28 + 1846875875*x^27 - 1007873658*x^26 - 5671315301*x^25 + 2797394685*x^24 + 12962901258*x^23 - 5730420663*x^22 - 21895326590*x^21 + 8590544711*x^20 + 27001558938*x^19 - 9297003415*x^18 - 23896748020*x^17 + 7121607714*x^16 + 14834334408*x^15 - 3755010451*x^14 - 6269987218*x^13 + 1310830451*x^12 + 1732796449*x^11 - 287529242*x^10 - 294221159*x^9 + 37041429*x^8 + 27505221*x^7 - 2599666*x^6 - 1134285*x^5 + 90620*x^4 + 12893*x^3 - 370*x^2 - 54*x - 1)
 
gp: K = bnfinit(x^43 - x^42 - 84*x^41 + 79*x^40 + 3160*x^39 - 2786*x^38 - 70521*x^37 + 58076*x^36 + 1042773*x^35 - 798942*x^34 - 10810701*x^33 + 7671648*x^32 + 81126975*x^31 - 53056499*x^30 - 448758019*x^29 + 268953093*x^28 + 1846875875*x^27 - 1007873658*x^26 - 5671315301*x^25 + 2797394685*x^24 + 12962901258*x^23 - 5730420663*x^22 - 21895326590*x^21 + 8590544711*x^20 + 27001558938*x^19 - 9297003415*x^18 - 23896748020*x^17 + 7121607714*x^16 + 14834334408*x^15 - 3755010451*x^14 - 6269987218*x^13 + 1310830451*x^12 + 1732796449*x^11 - 287529242*x^10 - 294221159*x^9 + 37041429*x^8 + 27505221*x^7 - 2599666*x^6 - 1134285*x^5 + 90620*x^4 + 12893*x^3 - 370*x^2 - 54*x - 1, 1)
 

Normalized defining polynomial

\( x^{43} - x^{42} - 84 x^{41} + 79 x^{40} + 3160 x^{39} - 2786 x^{38} - 70521 x^{37} + 58076 x^{36} + 1042773 x^{35} - 798942 x^{34} - 10810701 x^{33} + 7671648 x^{32} + 81126975 x^{31} - 53056499 x^{30} - 448758019 x^{29} + 268953093 x^{28} + 1846875875 x^{27} - 1007873658 x^{26} - 5671315301 x^{25} + 2797394685 x^{24} + 12962901258 x^{23} - 5730420663 x^{22} - 21895326590 x^{21} + 8590544711 x^{20} + 27001558938 x^{19} - 9297003415 x^{18} - 23896748020 x^{17} + 7121607714 x^{16} + 14834334408 x^{15} - 3755010451 x^{14} - 6269987218 x^{13} + 1310830451 x^{12} + 1732796449 x^{11} - 287529242 x^{10} - 294221159 x^{9} + 37041429 x^{8} + 27505221 x^{7} - 2599666 x^{6} - 1134285 x^{5} + 90620 x^{4} + 12893 x^{3} - 370 x^{2} - 54 x - 1 \)

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

Invariants

Degree:  $43$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[43, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9952594992767664919302480397055915291864099597331865326873171797085952964991380209309055259529=173^{42}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.46$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $173$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(173\)
Dirichlet character group:    $\lbrace$$\chi_{173}(1,·)$, $\chi_{173}(132,·)$, $\chi_{173}(133,·)$, $\chi_{173}(6,·)$, $\chi_{173}(135,·)$, $\chi_{173}(136,·)$, $\chi_{173}(10,·)$, $\chi_{173}(139,·)$, $\chi_{173}(140,·)$, $\chi_{173}(14,·)$, $\chi_{173}(16,·)$, $\chi_{173}(148,·)$, $\chi_{173}(149,·)$, $\chi_{173}(22,·)$, $\chi_{173}(23,·)$, $\chi_{173}(152,·)$, $\chi_{173}(29,·)$, $\chi_{173}(158,·)$, $\chi_{173}(160,·)$, $\chi_{173}(36,·)$, $\chi_{173}(169,·)$, $\chi_{173}(43,·)$, $\chi_{173}(47,·)$, $\chi_{173}(51,·)$, $\chi_{173}(52,·)$, $\chi_{173}(57,·)$, $\chi_{173}(60,·)$, $\chi_{173}(138,·)$, $\chi_{173}(81,·)$, $\chi_{173}(83,·)$, $\chi_{173}(84,·)$, $\chi_{173}(142,·)$, $\chi_{173}(164,·)$, $\chi_{173}(95,·)$, $\chi_{173}(96,·)$, $\chi_{173}(100,·)$, $\chi_{173}(106,·)$, $\chi_{173}(109,·)$, $\chi_{173}(117,·)$, $\chi_{173}(118,·)$, $\chi_{173}(119,·)$, $\chi_{173}(124,·)$, $\chi_{173}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{439} a^{40} - \frac{195}{439} a^{39} + \frac{16}{439} a^{38} - \frac{89}{439} a^{37} + \frac{194}{439} a^{36} - \frac{161}{439} a^{35} + \frac{42}{439} a^{34} - \frac{201}{439} a^{33} + \frac{152}{439} a^{32} - \frac{191}{439} a^{31} - \frac{35}{439} a^{30} + \frac{123}{439} a^{29} - \frac{153}{439} a^{28} - \frac{215}{439} a^{27} - \frac{140}{439} a^{26} + \frac{90}{439} a^{25} + \frac{84}{439} a^{24} + \frac{105}{439} a^{23} - \frac{49}{439} a^{22} + \frac{57}{439} a^{21} - \frac{173}{439} a^{20} + \frac{202}{439} a^{19} - \frac{27}{439} a^{18} - \frac{150}{439} a^{17} + \frac{79}{439} a^{16} + \frac{71}{439} a^{15} - \frac{33}{439} a^{14} + \frac{14}{439} a^{13} + \frac{134}{439} a^{12} + \frac{215}{439} a^{11} + \frac{76}{439} a^{10} + \frac{100}{439} a^{9} + \frac{187}{439} a^{8} + \frac{93}{439} a^{7} + \frac{143}{439} a^{6} - \frac{46}{439} a^{5} - \frac{79}{439} a^{4} + \frac{195}{439} a^{3} - \frac{139}{439} a^{2} - \frac{10}{439} a + \frac{66}{439}$, $\frac{1}{393729808938589} a^{41} + \frac{118184672889}{393729808938589} a^{40} - \frac{175419443235213}{393729808938589} a^{39} + \frac{140439653559240}{393729808938589} a^{38} + \frac{169328219003484}{393729808938589} a^{37} - \frac{17816338841554}{393729808938589} a^{36} - \frac{16660852818608}{393729808938589} a^{35} - \frac{92365897316206}{393729808938589} a^{34} - \frac{174069904687629}{393729808938589} a^{33} + \frac{74511950628496}{393729808938589} a^{32} + \frac{122586483809914}{393729808938589} a^{31} + \frac{158635369124277}{393729808938589} a^{30} - \frac{13848889877657}{393729808938589} a^{29} + \frac{140347423553191}{393729808938589} a^{28} - \frac{112968103160742}{393729808938589} a^{27} + \frac{153430105794647}{393729808938589} a^{26} - \frac{72736226978788}{393729808938589} a^{25} + \frac{168180617028682}{393729808938589} a^{24} - \frac{87525194788406}{393729808938589} a^{23} - \frac{187336522664663}{393729808938589} a^{22} + \frac{9276296544652}{393729808938589} a^{21} + \frac{2641420354443}{393729808938589} a^{20} + \frac{88456071253694}{393729808938589} a^{19} - \frac{175525679316799}{393729808938589} a^{18} - \frac{126527970694928}{393729808938589} a^{17} - \frac{126898159483620}{393729808938589} a^{16} - \frac{143143684221711}{393729808938589} a^{15} - \frac{145220588297682}{393729808938589} a^{14} - \frac{54014514099111}{393729808938589} a^{13} + \frac{134045097322187}{393729808938589} a^{12} + \frac{69303068603000}{393729808938589} a^{11} + \frac{32621135966434}{393729808938589} a^{10} + \frac{151443163986127}{393729808938589} a^{9} - \frac{115549570362463}{393729808938589} a^{8} + \frac{1028061930698}{393729808938589} a^{7} - \frac{25514004847786}{393729808938589} a^{6} + \frac{28879785465430}{393729808938589} a^{5} - \frac{178641133299543}{393729808938589} a^{4} + \frac{112500866236428}{393729808938589} a^{3} + \frac{179231976728897}{393729808938589} a^{2} + \frac{36082084950171}{393729808938589} a - \frac{179213920964281}{393729808938589}$, $\frac{1}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{42} + \frac{3145094173308403357721441223404124994569088485582213514291596483148235232813801458071690540101634970717899137608}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{41} + \frac{1949276166488548893790082092851662901969891912934606107451635567408323434824548759512695397081330130843159940344088739826353}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{40} + \frac{147071431840065500935828813586591264859408371039930611387734034816651080831224044420238411508358918263311061922659981017769412}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{39} + \frac{535329666712397300423618823212070290464497285435283705081067180848693613974876577774888431599249989010999547393488402339902995}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{38} - \frac{542003419476467109264076246449097794093373575623835090898128873269618569248890189534284350758278754704254935832397592818719540}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{37} - \frac{936235460464802298727377422987331659056198189625849346001884561755184282701660934532449984135195108365520692699758819167468958}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{36} - \frac{99360823042568089858823692991197825783707721250045721412184139048532361495046046045089135192872317759346336309058784706666783}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{35} - \frac{536829328023123369013972147459289664437160064355975996281360069339229459632492324447865678961318961653543988402085052786575334}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{34} - \frac{28835426018644609873490182670720430288972103303144115434453419298583359373582397713012289717651293196556087232651743518614599}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{33} + \frac{1012071220839733413864133069838940036387987852149998354434583580111809299454671101095988953876468356605772914641368967633053132}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{32} + \frac{1779369930710080976164804822787071503825464954095365365875246784358828847124887488001892955140667325714362574676044299844941359}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{31} - \frac{1144077421369563405378166068392049456976278394355400060007919258996737894072950360363810684561202699125841175818733803181478891}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{30} + \frac{1672778167329211884914059585428112734283065242956832931914378866465474661548767099365666128278544420093915710599728141982685831}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{29} - \frac{614164445967266249997482315315318760590903343071637689804489553575160460996804102047612741867443600610689620654783239942835665}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{28} + \frac{663734245407161728110822684097781105604178588502038380880839984645334508837847062842094219356709782659790672043541676576612997}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{27} - \frac{448500239088890476663809373841453309608569768153650679418691815826020408818915626667622730772587822464291304877523222993875694}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{26} + \frac{1432439735249145366681142776558224087655255006354598534532780980799359462562652765198867247804306593324923680821150169775967132}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{25} - \frac{1696078886928258319113217896222563453298355870073676831388964806808889070088985439235484888594753994866716026253284881866501634}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{24} - \frac{790605553496790343842468880369244400321734353153888073858284732781038393756464139929606387695552233639398874473067029931579247}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{23} - \frac{518855723270673521937687050641639167748714798860810540382799096769491307809385725625621298512951044784359071313543685780565453}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{22} + \frac{722639701423359547429332758925639870237988181855839764587905884615840298487665384774507082844681755429993115371989275377579898}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{21} - \frac{230598368278960297873757309304329859215454047018614702045096231402760169603448691030451139118584395557845790730576228520393380}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{20} - \frac{595962173291925776057180370932956733268856119690383690195672211921917672302982023579718583119551793527331152556180144278546785}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{19} - \frac{1546257159818638652023755244057111840727615645397142277828088339403540975877234531273696723368201889327854692243846654374625359}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{18} - \frac{1006728538766171198148091031948579180833735941583218039527585211186446582914480011329432248138707949435662045648725606975035006}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{17} + \frac{257688222097768300431066301024112834355868250663663509135653308500068589130034352431615940875582169896486657366161048102049957}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{16} - \frac{1405336094686755385612591169377939296904943118707308568305686439877844315956137737787677760334068620439977030310463040058032121}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{15} + \frac{945582400699845157248534222495119471663201367837005992395857336925821054830517062298716308096156404131904220604818308663178999}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{14} + \frac{484460274711693010663815928780068877916199531977572593754083437483439445330184068871939393666345199593663111806568795495328450}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{13} - \frac{1295402704803149288254248490432819148800979055384613883654142898649455881349144000177496028287432605660815546990122286147269039}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{12} - \frac{315051397993192437409870476499749140846858481913849670891664064336037212445263194709215346162813451511418328338239612158317250}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{11} + \frac{1147413739029087241545596409439132893772578284271887778353701587267084260443578430438635220453561076543821576257868732003358903}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{10} + \frac{1537228712174329195110212396614780678664308943246338013572109346654379043989704735119131418994961234341390005667479049806481480}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{9} - \frac{261283474076537810641033381089598743125534083209609068494222450272918168752781482062098136782187711484486158395271173072123403}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{8} - \frac{145066710432419834917642464274333915120532214014768182826539364474394631655641419904434503085040935717673192376655629498189715}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{7} + \frac{1281418934253633870878396331449372090090436345157777355698247899585060350920622961811164184601923907195974260399441998975709605}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{6} + \frac{655489270852382152610374291472715173259219017388406519768149243757856345518119038572431773382320410238903356342617808189361922}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{5} + \frac{1371519732477809663200594864705011619348325389230579865754962634011740752331543071695220910296998494703493296874609722817429272}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{4} + \frac{6986894180016222007380247574817876128825107023736715870374255392324457994161238574448496956151478689715838656189656086376246}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{3} + \frac{691832017146784140405213281327738447908912031221926848574396278942471372453628019352531880457746728507937636749521754812841340}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a^{2} - \frac{1197840035011039117592827374619305649238214615542534189077633301229939108672721412404346904756510285010575434084751221442053204}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759} a + \frac{367342496763641701685607229708401429764587887441747787249721699946324265506660794668927452997264269308590741641730804386875596}{3573753624332479486386642290095414066635118253584413695758380524067450619801296522039897861619953372817088562705395528461888759}$

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:  $42$
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:  \( 2657880348049265700000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{43}$ (as 43T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 43
The 43 conjugacy class representatives for $C_{43}$
Character table for $C_{43}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$ $43$

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
173Data not computed