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Properties

Label 46.46.1559562745...4357.1
Degree $46$
Signature $[46, 0]$
Discriminant $13^{23}\cdot 47^{44}$
Root discriminant $143.34$
Ramified primes $13, 47$
Class number Not computed
Class group Not computed
Galois group $C_{46}$ (as 46T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-400721, 9031748, -44784143, -387991071, 5444167373, -23274902617, 16546070228, 216958911989, -822188957900, 719437437556, 2524580197417, -7356504751997, 3424903604271, 14052337420836, -22787259295206, -2181540751435, 34519826963183, -22251983955427, -21465428316192, 31269803408789, 1046437412802, -20645533933596, 7178933729591, 7582856629140, -5285056712013, -1355370040843, 2018290007358, -66786211526, -480319603281, 103336700236, 72586582734, -28571535040, -6237187098, 4479212294, 100002271, -443216122, 44109968, 26988590, -5657221, -834537, 335613, -1825, -9950, 959, 96, -21, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 21*x^45 + 96*x^44 + 959*x^43 - 9950*x^42 - 1825*x^41 + 335613*x^40 - 834537*x^39 - 5657221*x^38 + 26988590*x^37 + 44109968*x^36 - 443216122*x^35 + 100002271*x^34 + 4479212294*x^33 - 6237187098*x^32 - 28571535040*x^31 + 72586582734*x^30 + 103336700236*x^29 - 480319603281*x^28 - 66786211526*x^27 + 2018290007358*x^26 - 1355370040843*x^25 - 5285056712013*x^24 + 7582856629140*x^23 + 7178933729591*x^22 - 20645533933596*x^21 + 1046437412802*x^20 + 31269803408789*x^19 - 21465428316192*x^18 - 22251983955427*x^17 + 34519826963183*x^16 - 2181540751435*x^15 - 22787259295206*x^14 + 14052337420836*x^13 + 3424903604271*x^12 - 7356504751997*x^11 + 2524580197417*x^10 + 719437437556*x^9 - 822188957900*x^8 + 216958911989*x^7 + 16546070228*x^6 - 23274902617*x^5 + 5444167373*x^4 - 387991071*x^3 - 44784143*x^2 + 9031748*x - 400721)
 
gp: K = bnfinit(x^46 - 21*x^45 + 96*x^44 + 959*x^43 - 9950*x^42 - 1825*x^41 + 335613*x^40 - 834537*x^39 - 5657221*x^38 + 26988590*x^37 + 44109968*x^36 - 443216122*x^35 + 100002271*x^34 + 4479212294*x^33 - 6237187098*x^32 - 28571535040*x^31 + 72586582734*x^30 + 103336700236*x^29 - 480319603281*x^28 - 66786211526*x^27 + 2018290007358*x^26 - 1355370040843*x^25 - 5285056712013*x^24 + 7582856629140*x^23 + 7178933729591*x^22 - 20645533933596*x^21 + 1046437412802*x^20 + 31269803408789*x^19 - 21465428316192*x^18 - 22251983955427*x^17 + 34519826963183*x^16 - 2181540751435*x^15 - 22787259295206*x^14 + 14052337420836*x^13 + 3424903604271*x^12 - 7356504751997*x^11 + 2524580197417*x^10 + 719437437556*x^9 - 822188957900*x^8 + 216958911989*x^7 + 16546070228*x^6 - 23274902617*x^5 + 5444167373*x^4 - 387991071*x^3 - 44784143*x^2 + 9031748*x - 400721, 1)
 

Normalized defining polynomial

\( x^{46} - 21 x^{45} + 96 x^{44} + 959 x^{43} - 9950 x^{42} - 1825 x^{41} + 335613 x^{40} - 834537 x^{39} - 5657221 x^{38} + 26988590 x^{37} + 44109968 x^{36} - 443216122 x^{35} + 100002271 x^{34} + 4479212294 x^{33} - 6237187098 x^{32} - 28571535040 x^{31} + 72586582734 x^{30} + 103336700236 x^{29} - 480319603281 x^{28} - 66786211526 x^{27} + 2018290007358 x^{26} - 1355370040843 x^{25} - 5285056712013 x^{24} + 7582856629140 x^{23} + 7178933729591 x^{22} - 20645533933596 x^{21} + 1046437412802 x^{20} + 31269803408789 x^{19} - 21465428316192 x^{18} - 22251983955427 x^{17} + 34519826963183 x^{16} - 2181540751435 x^{15} - 22787259295206 x^{14} + 14052337420836 x^{13} + 3424903604271 x^{12} - 7356504751997 x^{11} + 2524580197417 x^{10} + 719437437556 x^{9} - 822188957900 x^{8} + 216958911989 x^{7} + 16546070228 x^{6} - 23274902617 x^{5} + 5444167373 x^{4} - 387991071 x^{3} - 44784143 x^{2} + 9031748 x - 400721 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[46, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1559562745220946355455174068757250546262830593902488581782176430066187304634955035230281728558464357=13^{23}\cdot 47^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.34$
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:  $13, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(611=13\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{611}(1,·)$, $\chi_{611}(259,·)$, $\chi_{611}(519,·)$, $\chi_{611}(324,·)$, $\chi_{611}(521,·)$, $\chi_{611}(12,·)$, $\chi_{611}(14,·)$, $\chi_{611}(272,·)$, $\chi_{611}(131,·)$, $\chi_{611}(404,·)$, $\chi_{611}(534,·)$, $\chi_{611}(25,·)$, $\chi_{611}(27,·)$, $\chi_{611}(285,·)$, $\chi_{611}(545,·)$, $\chi_{611}(155,·)$, $\chi_{611}(168,·)$, $\chi_{611}(363,·)$, $\chi_{611}(298,·)$, $\chi_{611}(300,·)$, $\chi_{611}(430,·)$, $\chi_{611}(157,·)$, $\chi_{611}(51,·)$, $\chi_{611}(53,·)$, $\chi_{611}(183,·)$, $\chi_{611}(441,·)$, $\chi_{611}(571,·)$, $\chi_{611}(573,·)$, $\chi_{611}(64,·)$, $\chi_{611}(194,·)$, $\chi_{611}(196,·)$, $\chi_{611}(79,·)$, $\chi_{611}(337,·)$, $\chi_{611}(378,·)$, $\chi_{611}(142,·)$, $\chi_{611}(220,·)$, $\chi_{611}(222,·)$, $\chi_{611}(350,·)$, $\chi_{611}(144,·)$, $\chi_{611}(482,·)$, $\chi_{611}(209,·)$, $\chi_{611}(103,·)$, $\chi_{611}(365,·)$, $\chi_{611}(495,·)$, $\chi_{611}(118,·)$, $\chi_{611}(506,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $\frac{1}{283} a^{43} - \frac{68}{283} a^{42} + \frac{22}{283} a^{41} + \frac{111}{283} a^{40} - \frac{66}{283} a^{39} + \frac{129}{283} a^{38} - \frac{99}{283} a^{37} + \frac{117}{283} a^{36} - \frac{99}{283} a^{35} + \frac{63}{283} a^{34} + \frac{61}{283} a^{33} - \frac{40}{283} a^{32} + \frac{138}{283} a^{31} + \frac{129}{283} a^{30} + \frac{91}{283} a^{29} + \frac{80}{283} a^{28} + \frac{59}{283} a^{27} + \frac{104}{283} a^{26} + \frac{58}{283} a^{25} + \frac{115}{283} a^{24} + \frac{55}{283} a^{23} - \frac{104}{283} a^{22} + \frac{129}{283} a^{21} + \frac{85}{283} a^{20} + \frac{91}{283} a^{19} - \frac{118}{283} a^{18} - \frac{32}{283} a^{17} - \frac{79}{283} a^{16} - \frac{24}{283} a^{15} - \frac{128}{283} a^{14} - \frac{19}{283} a^{13} + \frac{114}{283} a^{12} - \frac{71}{283} a^{11} - \frac{15}{283} a^{10} - \frac{139}{283} a^{9} + \frac{6}{283} a^{8} - \frac{37}{283} a^{7} + \frac{84}{283} a^{6} - \frac{46}{283} a^{5} - \frac{141}{283} a^{4} - \frac{128}{283} a^{3} + \frac{120}{283} a^{2} + \frac{25}{283} a - \frac{89}{283}$, $\frac{1}{1595837} a^{44} + \frac{2259}{1595837} a^{43} + \frac{694748}{1595837} a^{42} - \frac{762886}{1595837} a^{41} + \frac{675939}{1595837} a^{40} - \frac{354100}{1595837} a^{39} - \frac{482977}{1595837} a^{38} - \frac{511558}{1595837} a^{37} + \frac{288291}{1595837} a^{36} + \frac{351821}{1595837} a^{35} + \frac{627762}{1595837} a^{34} - \frac{115906}{1595837} a^{33} + \frac{345142}{1595837} a^{32} - \frac{235972}{1595837} a^{31} - \frac{774843}{1595837} a^{30} - \frac{122952}{1595837} a^{29} + \frac{8495}{1595837} a^{28} - \frac{651890}{1595837} a^{27} + \frac{325268}{1595837} a^{26} - \frac{723541}{1595837} a^{25} - \frac{445783}{1595837} a^{24} + \frac{499743}{1595837} a^{23} + \frac{638817}{1595837} a^{22} + \frac{196690}{1595837} a^{21} + \frac{794450}{1595837} a^{20} - \frac{121452}{1595837} a^{19} + \frac{31305}{1595837} a^{18} - \frac{249437}{1595837} a^{17} - \frac{797684}{1595837} a^{16} + \frac{331168}{1595837} a^{15} + \frac{432831}{1595837} a^{14} + \frac{56083}{1595837} a^{13} + \frac{71918}{1595837} a^{12} + \frac{8247}{1595837} a^{11} + \frac{226448}{1595837} a^{10} - \frac{131290}{1595837} a^{9} + \frac{147218}{1595837} a^{8} - \frac{3096}{1595837} a^{7} + \frac{378240}{1595837} a^{6} - \frac{126993}{1595837} a^{5} - \frac{570483}{1595837} a^{4} - \frac{787326}{1595837} a^{3} - \frac{90050}{1595837} a^{2} - \frac{360754}{1595837} a + \frac{151741}{1595837}$, $\frac{1}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{45} - \frac{369007065415247571938268540474647076314867633806970837944709664911609985945951870704604434521555051855164983662252241543873278232435381656639033420006572509504875478978114816770080414761594493425917281}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{44} - \frac{21031794185694186135952903554580892048854220538934784975552560535679069619110448539747247840453876238194983653310518269675892340706839884485693179859429779191646980132640904024430187610594349024865899390758}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{43} - \frac{308316950364092950654830735040016373874078664159350396647464059930700695929042938333922628691604231949803390677865047260492468042952184175969951662149932069371485023516131588080766194733175321963115806845601}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{42} - \frac{8835097457126861500462664418281361875073362308035742760685009556060914914106307354478255634598157683294074559779881954682147799515826592227915549609811152905798002585292432481511033507562343307186896980710383}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{41} - \frac{3697001983924175170816650492709034955008107018646301043624114853253637189265557274813042134811634637513921794156521427195253102035210993756448348458826815895953053069625574715788835041487076222732307533507133}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{40} + \frac{2902118634457376020864657130431978418734204732806844729204897240712476581451456880936132825846589149585380139419201714675041185494002079167285585556243101113495639310051931507723096643962756261763063812596283}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{39} + \frac{6306852605533173605638306450772619337796192659808379355854621105175920219282541471407521775722036535262490612517325688089349166629139116686037047486643769574506936738414953618967089068588719394888466455920283}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{38} + \frac{4175468277382455193207525990433109085575534720086483187078421906429717205737338015118895515498668053076158904028442580209547809268465993159726747282954561046993167186181799721427916986467455905612865206096688}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{37} - \frac{3098606760376046508934925423399017188995960725866082808085334465276862871074569021808336118772005236680767269947929882069891647700814085381243452418571202459000698102473949042286284649788138741466341123305236}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{36} + \frac{480045089442479993328637039251819152172880688254406588212224214119245919288380192788534414893312665027722197719830836533931600157869077623852172504831192538123589054455252492558945659791101906969393855405616}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{35} + \frac{6940536670105681005142447597869234533059945251499611389957215076258958399510253984308097521000089692948964431357165248704032856701833471368925608720188941138565242388402713965976917843240050422157759152936596}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{34} + \frac{8495719753647333242887462001675431509266505503282365778204398850047983693759631249699236605167266967067193849809664034352519751176902775592471282764502645543645284905156232091698504339408181495982660193149310}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{33} + \frac{3967287985227372867754542135127998347064677560686000568186731205725041770232667833269788999685578867544109038744008193417094292710525036719715163954736553091697052725230415942916643511081852391139065599692070}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{32} + \frac{3089496636107521196349822040489210012055583618189121285543451842321696160797709661541448593904523787450532357581341552074914443381646602290716160673032424958714376649278921039224044704796373622392305191433767}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{31} - \frac{628233141241214246366852713056037482285920603677336505440538890711925211281291400148360059327365324341203024785759328261007418088608244741968185124063940785614837853593376142855175122547052505790491972212767}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{30} - \frac{3560501124074519225395500670919736480466730964510335976112576094021947230744403361486477146256781136943207727105151469165629046944558371320370814835453823038199507995668919031247966930066649280505583332557081}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{29} - \frac{5875392148638965211854316880307968505642775622139398366686629334934710469329692213123611913523701598005931750655182119071764647820399057791417653858947719161750803433908505799069853992737533239870710608073073}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{28} + \frac{6186606999344938881088573219237156163252649752788559654093492715483241537337990587636061279032187125304675164853027044384356993645959880272752350499862864423446521184445512199961066539702647920852639618278690}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{27} - \frac{8611010682817601640071110735566044312874850752724349246029556032197115648600618344111282267170803344436013295730727084736629135720133483744112900124815750510941163367779685410557824664050145704719034437786068}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{26} - \frac{2218649245752483772261605599066902734982843672955394035869265473180181967224607315365508546976889877009959317399621526119695405810828962066568306650937277583754787782297995802646732227289293290864466624293786}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{25} - \frac{8232191066059746875664839439474170412254778733763519677581289770744903331408681763866027586125999049284715727691131843756781220772412160251710791920368453921042993184400615930538077568373630114174222943991150}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{24} - \frac{3761823800187407678070064860510555035170529006462060953906361101681255698129322928698725016401442721814453227081336795448320632714339177474233755254717761309151968922975295578988876876549940388731328277254309}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{23} + \frac{5309032536593783675788231165770593278691205530475697777762716005058283612147358909044354025171068393301205115347967122003561389574274383857812297269993342544740435482424346213208063917007778590844209543177215}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{22} - \frac{6105033727666115236039511232292706164918931843055058748751563198628530916306823180055278282964722049515376666146357418444242752777094101093959674100621861102117279471208739197558341681893029673695340326707519}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{21} + \frac{6892431038758185568017037479613408865954171169333730069557212463347573483962538316331410875197333430776489828252756425447827625344269924296839149715474813519324964107654501569002641916555634036541644997994208}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{20} + \frac{903976564994383739866952084522649722955360647973117865882249751415941062424648381078608681229880245615025424020122264464314048582700152412953050265965554406954806248792703305511482761091345103166718716318519}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{19} - \frac{3636902735750521053959455927175080305508453948342808374049662796086483738989456320133752095622441468800637105188082447249185441726816898103332063400067430514138185337993492017722835473449378400838945940513170}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{18} + \frac{5045162612122650725335498167514002670553118544844147450706065091890987569636982345642095910248586749282559789760554184016636776428250641997860142539423349666372460282346562058734044761230080259988264653642589}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{17} + \frac{7813936458606992955994342198442543380047339473272069101765412019793934586005447989924638915904389994344137138631029221333868017727155595235499239853506801203843543913320582516168217199290066945262591023537434}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{16} + \frac{296286961543002358036398832015836307281993865783047662333749828209067301019073163640219977703974363491222379237142714668552949207643929619395202736823180670682785377570623355323626481426850539220503420156065}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{15} - \frac{5423751006905951534238411739296474755288013142222664057523788505397099616585317399685252005495048296902638925730521788523250658691166779561678044588373246515770999924383459112046946920859321991896532674762889}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{14} - \frac{4818999404911322188670681904001586878728653595225891878071521556011616569714514624366255531365767074918528045770511183925549525993837454518378103528631250495830635785918014926909320162582362086003228857947821}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{13} - \frac{5525513485622349167484062372563272100608532287660612624243794708460180502280632638509472498440515124173622798840406745153190907461282864182646992628600503884007942957539707684556259890902991329622104757412007}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{12} + \frac{6827087696231164083669743405436414161294362807941010883634552981152922216523933188251201347747405118226551280742394387429594261609423383113560528856082086700857154639385650853144225799998505997806899056493244}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{11} + \frac{3607945578771932449403172675899539543349496823699246173651800009654265324917693964025623977781676679973490666390878459773843275118404057960734793239824004578848494271616078703002899216264422664779839634758776}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{10} + \frac{3652387980311554088248543714975703742882523298817056808685876834994192485977510867414733627575473366024568010612132497079611527580651797676068376887994865677946673069488248471771678561852358846699629713866597}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{9} + \frac{464220003493346799729980927866337445589921520106196692162363444395348133485873242551540968035557472119795535104938598463836660659905074522209554587379615406127005487087976291488626586851911074540939699032154}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{8} + \frac{3538466603866124080655721551229716751668637073517248062991513947094458749928154899004265217036840193736655295036585746820228628688498722546250090198218581303905471353058267384273851821258739963381263183974315}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{7} - \frac{8109672450883702131816087410556840256467615395783201171551411803213939296639337556749768125622382161642598191475137563545268801265125821180110857767481780431053337948203673431451160593624937555869355961393479}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{6} + \frac{8427730061384992112848073323533117401494181612896264074870232417233798122930513005518550123279185199335488727875539959000223512234054224917219645724567848278198783657654257734261744200597412364087516605977637}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{5} - \frac{1605703866093918516189758470156053893709215683227857157964525496429426904102373330099599195845086747519522269958310581181055444756869667094673395668199270067572603726518282224192605993961173725534797458037408}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{4} - \frac{8252771381604349737900687456916224981239234778238459180910716840311524456103469449942472627631131921630464451855331363679770718654035678263862355015097499492388517646299184496787729675771333330144173474481261}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{3} - \frac{5480231697747267255756060874131789170852320711391942644219416888157624048217876843607719012977156908562713302025171675122930192473218939405808233651009471875275524767559357572008378533183850197965868565444046}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a^{2} - \frac{4036401986429215195771558367284069819394478383741177786620721761577280472412660629072467304491505759305405695544708291890816652515916325729885452337510933727066934686862433490810034096340443099227092526703210}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651} a - \frac{4341433227171332463864199138581203420145179321997090588964027043048362042610931417521325969227948486513449731493436909433075534308519575978656557741311979555641259857719457378963043516580619381359262022900737}{19522653533790080223671100499860722222117276190195287609941931596581025151702128921204443121974104482334255241109466386905685140536096440164082909770375277459522440650079339331526007094638222015248283208238651}$

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:  $45$
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

$C_{46}$ (as 46T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{47})^+\)

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 $46$ $23^{2}$ $46$ $46$ $46$ R $23^{2}$ $46$ $23^{2}$ $23^{2}$ $46$ $46$ $46$ $23^{2}$ R $23^{2}$ $46$

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
47Data not computed