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Properties

Label 46.0.73299449025...4779.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,13^{23}\cdot 47^{45}$
Root discriminant $155.85$
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![106127132605244053, -106122707875839184, 106122707875839184, -105987016174089868, 105987016174089868, -104743175574721138, 104743175574721138, -99353199644123308, 99353199644123308, -85878259817628733, 85878259817628733, -64155023491279903, 64155023491279903, -39925259896506208, 39925259896506208, -20310689367403693, 20310689367403693, -8388107281086478, 8388107281086478, -2810291100353278, 2810291100353278, -765091834084438, 765091834084438, -169585855368478, 169585855368478, -30634460334754, 30634460334754, -4506847593370, 4506847593370, -538367825590, 538367825590, -51909015346, 51909015346, -4000193125, 4000193125, -242638441, 242638441, -11317507, 11317507, -391417, 391417, -9448, 9448, -142, 142, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 + 142*x^44 - 142*x^43 + 9448*x^42 - 9448*x^41 + 391417*x^40 - 391417*x^39 + 11317507*x^38 - 11317507*x^37 + 242638441*x^36 - 242638441*x^35 + 4000193125*x^34 - 4000193125*x^33 + 51909015346*x^32 - 51909015346*x^31 + 538367825590*x^30 - 538367825590*x^29 + 4506847593370*x^28 - 4506847593370*x^27 + 30634460334754*x^26 - 30634460334754*x^25 + 169585855368478*x^24 - 169585855368478*x^23 + 765091834084438*x^22 - 765091834084438*x^21 + 2810291100353278*x^20 - 2810291100353278*x^19 + 8388107281086478*x^18 - 8388107281086478*x^17 + 20310689367403693*x^16 - 20310689367403693*x^15 + 39925259896506208*x^14 - 39925259896506208*x^13 + 64155023491279903*x^12 - 64155023491279903*x^11 + 85878259817628733*x^10 - 85878259817628733*x^9 + 99353199644123308*x^8 - 99353199644123308*x^7 + 104743175574721138*x^6 - 104743175574721138*x^5 + 105987016174089868*x^4 - 105987016174089868*x^3 + 106122707875839184*x^2 - 106122707875839184*x + 106127132605244053)
 
gp: K = bnfinit(x^46 - x^45 + 142*x^44 - 142*x^43 + 9448*x^42 - 9448*x^41 + 391417*x^40 - 391417*x^39 + 11317507*x^38 - 11317507*x^37 + 242638441*x^36 - 242638441*x^35 + 4000193125*x^34 - 4000193125*x^33 + 51909015346*x^32 - 51909015346*x^31 + 538367825590*x^30 - 538367825590*x^29 + 4506847593370*x^28 - 4506847593370*x^27 + 30634460334754*x^26 - 30634460334754*x^25 + 169585855368478*x^24 - 169585855368478*x^23 + 765091834084438*x^22 - 765091834084438*x^21 + 2810291100353278*x^20 - 2810291100353278*x^19 + 8388107281086478*x^18 - 8388107281086478*x^17 + 20310689367403693*x^16 - 20310689367403693*x^15 + 39925259896506208*x^14 - 39925259896506208*x^13 + 64155023491279903*x^12 - 64155023491279903*x^11 + 85878259817628733*x^10 - 85878259817628733*x^9 + 99353199644123308*x^8 - 99353199644123308*x^7 + 104743175574721138*x^6 - 104743175574721138*x^5 + 105987016174089868*x^4 - 105987016174089868*x^3 + 106122707875839184*x^2 - 106122707875839184*x + 106127132605244053, 1)
 

Normalized defining polynomial

\( x^{46} - x^{45} + 142 x^{44} - 142 x^{43} + 9448 x^{42} - 9448 x^{41} + 391417 x^{40} - 391417 x^{39} + 11317507 x^{38} - 11317507 x^{37} + 242638441 x^{36} - 242638441 x^{35} + 4000193125 x^{34} - 4000193125 x^{33} + 51909015346 x^{32} - 51909015346 x^{31} + 538367825590 x^{30} - 538367825590 x^{29} + 4506847593370 x^{28} - 4506847593370 x^{27} + 30634460334754 x^{26} - 30634460334754 x^{25} + 169585855368478 x^{24} - 169585855368478 x^{23} + 765091834084438 x^{22} - 765091834084438 x^{21} + 2810291100353278 x^{20} - 2810291100353278 x^{19} + 8388107281086478 x^{18} - 8388107281086478 x^{17} + 20310689367403693 x^{16} - 20310689367403693 x^{15} + 39925259896506208 x^{14} - 39925259896506208 x^{13} + 64155023491279903 x^{12} - 64155023491279903 x^{11} + 85878259817628733 x^{10} - 85878259817628733 x^{9} + 99353199644123308 x^{8} - 99353199644123308 x^{7} + 104743175574721138 x^{6} - 104743175574721138 x^{5} + 105987016174089868 x^{4} - 105987016174089868 x^{3} + 106122707875839184 x^{2} - 106122707875839184 x + 106127132605244053 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 23]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-73299449025384478706393181231590775674353037913416963343762292213110803317842886655823241242247824779=-\,13^{23}\cdot 47^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.85$
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$$\chi_{611}(1,·)$, $\chi_{611}(131,·)$, $\chi_{611}(389,·)$, $\chi_{611}(129,·)$, $\chi_{611}(521,·)$, $\chi_{611}(14,·)$, $\chi_{611}(365,·)$, $\chi_{611}(144,·)$, $\chi_{611}(402,·)$, $\chi_{611}(532,·)$, $\chi_{611}(558,·)$, $\chi_{611}(534,·)$, $\chi_{611}(27,·)$, $\chi_{611}(157,·)$, $\chi_{611}(415,·)$, $\chi_{611}(38,·)$, $\chi_{611}(300,·)$, $\chi_{611}(430,·)$, $\chi_{611}(53,·)$, $\chi_{611}(311,·)$, $\chi_{611}(428,·)$, $\chi_{611}(573,·)$, $\chi_{611}(181,·)$, $\chi_{611}(196,·)$, $\chi_{611}(118,·)$, $\chi_{611}(454,·)$, $\chi_{611}(584,·)$, $\chi_{611}(183,·)$, $\chi_{611}(77,·)$, $\chi_{611}(79,·)$, $\chi_{611}(209,·)$, $\chi_{611}(467,·)$, $\chi_{611}(597,·)$, $\chi_{611}(90,·)$, $\chi_{611}(207,·)$, $\chi_{611}(222,·)$, $\chi_{611}(480,·)$, $\chi_{611}(482,·)$, $\chi_{611}(233,·)$, $\chi_{611}(610,·)$, $\chi_{611}(493,·)$, $\chi_{611}(495,·)$, $\chi_{611}(116,·)$, $\chi_{611}(246,·)$, $\chi_{611}(404,·)$, $\chi_{611}(378,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{29434370640613867} a^{24} + \frac{8734687734666649}{29434370640613867} a^{23} + \frac{72}{29434370640613867} a^{22} + \frac{14006040879721441}{29434370640613867} a^{21} + \frac{2268}{29434370640613867} a^{20} + \frac{8100037423049092}{29434370640613867} a^{19} + \frac{41040}{29434370640613867} a^{18} + \frac{3112534987315685}{29434370640613867} a^{17} + \frac{470934}{29434370640613867} a^{16} - \frac{2212283543248379}{29434370640613867} a^{15} + \frac{3569184}{29434370640613867} a^{14} + \frac{13948385837875214}{29434370640613867} a^{13} + \frac{18044208}{29434370640613867} a^{12} - \frac{1795999352297662}{29434370640613867} a^{11} + \frac{60046272}{29434370640613867} a^{10} + \frac{8374173991550827}{29434370640613867} a^{9} + \frac{126660105}{29434370640613867} a^{8} - \frac{8473983327699605}{29434370640613867} a^{7} + \frac{157621464}{29434370640613867} a^{6} - \frac{8473983327699605}{29434370640613867} a^{5} + \frac{101328084}{29434370640613867} a^{4} - \frac{1646886871198751}{29434370640613867} a^{3} + \frac{25509168}{29434370640613867} a^{2} - \frac{4238353297065357}{29434370640613867} a + \frac{1062882}{29434370640613867}$, $\frac{1}{29434370640613867} a^{25} + \frac{75}{29434370640613867} a^{23} + \frac{3230307436613920}{29434370640613867} a^{22} + \frac{2475}{29434370640613867} a^{21} + \frac{7159696332221651}{29434370640613867} a^{20} + \frac{47250}{29434370640613867} a^{19} + \frac{12727936304326918}{29434370640613867} a^{18} + \frac{577125}{29434370640613867} a^{17} - \frac{10348895259015295}{29434370640613867} a^{16} + \frac{4709340}{29434370640613867} a^{15} + \frac{14248529972302518}{29434370640613867} a^{14} + \frac{26025300}{29434370640613867} a^{13} - \frac{6819575259496040}{29434370640613867} a^{12} + \frac{96665400}{29434370640613867} a^{11} + \frac{1303622695192702}{29434370640613867} a^{10} + \frac{234555750}{29434370640613867} a^{9} - \frac{852737446735033}{29434370640613867} a^{8} + \frac{351833625}{29434370640613867} a^{7} + \frac{13438079150204384}{29434370640613867} a^{6} + \frac{295540245}{29434370640613867} a^{5} + \frac{10336983961695680}{29434370640613867} a^{4} + \frac{115145550}{29434370640613867} a^{3} + \frac{3101095188508704}{29434370640613867} a^{2} + \frac{13286025}{29434370640613867} a + \frac{11343699343795786}{29434370640613867}$, $\frac{1}{29434370640613867} a^{26} - \frac{4315118569879681}{29434370640613867} a^{23} - \frac{2925}{29434370640613867} a^{22} - \frac{13090397225401079}{29434370640613867} a^{21} - \frac{122850}{29434370640613867} a^{20} - \frac{6087457612077642}{29434370640613867} a^{19} - \frac{2500875}{29434370640613867} a^{18} - \frac{8314054182780734}{29434370640613867} a^{17} - \frac{30610710}{29434370640613867} a^{16} + \frac{3563571872247741}{29434370640613867} a^{15} - \frac{241663500}{29434370640613867} a^{14} + \frac{6688829961962122}{29434370640613867} a^{13} - \frac{1256650200}{29434370640613867} a^{12} - \frac{11168279085551983}{29434370640613867} a^{11} - \frac{4268914650}{29434370640613867} a^{10} - \frac{10794003360155851}{29434370640613867} a^{9} - \frac{9147674250}{29434370640613867} a^{8} + \frac{1430674634169685}{29434370640613867} a^{7} - \frac{11526069555}{29434370640613867} a^{6} - \frac{1670420554339019}{29434370640613867} a^{5} - \frac{7484460750}{29434370640613867} a^{4} + \frac{8880127965959561}{29434370640613867} a^{3} - \frac{1899901575}{29434370640613867} a^{2} + \frac{5442119576945024}{29434370640613867} a - \frac{79716150}{29434370640613867}$, $\frac{1}{29434370640613867} a^{27} - \frac{3159}{29434370640613867} a^{23} + \frac{3254433399797283}{29434370640613867} a^{22} - \frac{138996}{29434370640613867} a^{21} + \frac{8390406191235022}{29434370640613867} a^{20} - \frac{2985255}{29434370640613867} a^{19} + \frac{6978279746303634}{29434370640613867} a^{18} - \frac{38893608}{29434370640613867} a^{17} - \frac{9336868391437885}{29434370640613867} a^{16} - \frac{330595668}{29434370640613867} a^{15} + \frac{12711958117246277}{29434370640613867} a^{14} - \frac{1879175376}{29434370640613867} a^{13} - \frac{1976176652058770}{29434370640613867} a^{12} - \frac{7125206634}{29434370640613867} a^{11} + \frac{10890326224728293}{29434370640613867} a^{10} - \frac{17563534560}{29434370640613867} a^{9} - \frac{4339091268789591}{29434370640613867} a^{8} - \frac{26674618113}{29434370640613867} a^{7} - \frac{3617796675245875}{29434370640613867} a^{6} - \frac{22633009308}{29434370640613867} a^{5} - \frac{13396307140776180}{29434370640613867} a^{4} - \frac{8891539371}{29434370640613867} a^{3} - \frac{7781820126168261}{29434370640613867} a^{2} - \frac{1033121304}{29434370640613867} a - \frac{1777429597655298}{29434370640613867}$, $\frac{1}{29434370640613867} a^{28} - \frac{13306673684065772}{29434370640613867} a^{23} + \frac{88452}{29434370640613867} a^{22} + \frac{13614472388625040}{29434370640613867} a^{21} + \frac{4179357}{29434370640613867} a^{20} - \frac{12905958175679028}{29434370640613867} a^{19} + \frac{90751752}{29434370640613867} a^{18} - \frac{7918637426220548}{29434370640613867} a^{17} + \frac{1157084838}{29434370640613867} a^{16} + \frac{54086821103495}{29434370640613867} a^{15} + \frac{9395876880}{29434370640613867} a^{14} - \frac{2278163803216643}{29434370640613867} a^{13} + \frac{49876446438}{29434370640613867} a^{12} - \frac{11272464685723501}{29434370640613867} a^{11} + \frac{172122638688}{29434370640613867} a^{10} - \frac{11822657871593531}{29434370640613867} a^{9} + \frac{373444653582}{29434370640613867} a^{8} + \frac{12346154080320900}{29434370640613867} a^{7} + \frac{475293195468}{29434370640613867} a^{6} + \frac{2567643614790595}{29434370640613867} a^{5} + \frac{311203877985}{29434370640613867} a^{4} - \frac{413842854368211}{29434370640613867} a^{3} + \frac{79550340408}{29434370640613867} a^{2} + \frac{1903146452191424}{29434370640613867} a + \frac{3357644238}{29434370640613867}$, $\frac{1}{29434370640613867} a^{29} + \frac{98658}{29434370640613867} a^{23} + \frac{360746501103013}{29434370640613867} a^{22} + \frac{4883571}{29434370640613867} a^{21} - \frac{3599949343721807}{29434370640613867} a^{20} + \frac{111878172}{29434370640613867} a^{19} + \frac{2090861323987881}{29434370640613867} a^{18} + \frac{1518346620}{29434370640613867} a^{17} - \frac{12390568040909757}{29434370640613867} a^{16} + \frac{13274687592}{29434370640613867} a^{15} - \frac{11390790903013780}{29434370640613867} a^{14} + \frac{77028121422}{29434370640613867} a^{13} - \frac{108358777948730}{29434370640613867} a^{12} + \frac{296700912144}{29434370640613867} a^{11} + \frac{11477015940208425}{29434370640613867} a^{10} + \frac{740505637872}{29434370640613867} a^{9} + \frac{8144106740978017}{29434370640613867} a^{8} + \frac{1136002967190}{29434370640613867} a^{7} - \frac{9856639230763938}{29434370640613867} a^{6} + \frac{971913649707}{29434370640613867} a^{5} + \frac{2111393402280256}{29434370640613867} a^{4} + \frac{384493311972}{29434370640613867} a^{3} + \frac{10360591134587730}{29434370640613867} a^{2} + \frac{44940776724}{29434370640613867} a + \frac{2805257748484335}{29434370640613867}$, $\frac{1}{29434370640613867} a^{30} + \frac{3607465011030130}{29434370640613867} a^{23} - \frac{2219805}{29434370640613867} a^{22} + \frac{14383033356952197}{29434370640613867} a^{21} - \frac{111878172}{29434370640613867} a^{20} + \frac{11761670813158395}{29434370640613867} a^{19} - \frac{2530577700}{29434370640613867} a^{18} - \frac{78453107286076}{29434370640613867} a^{17} - \frac{33186718980}{29434370640613867} a^{16} - \frac{7779281256262203}{29434370640613867} a^{15} - \frac{275100433650}{29434370640613867} a^{14} - \frac{4262161891301558}{29434370640613867} a^{13} - \frac{1483504560720}{29434370640613867} a^{12} + \frac{6269858427466681}{29434370640613867} a^{11} - \frac{5183539465104}{29434370640613867} a^{10} - \frac{7198410930493193}{29434370640613867} a^{9} - \frac{11360029671900}{29434370640613867} a^{8} - \frac{8038800398798249}{29434370640613867} a^{7} - \frac{14578704745605}{29434370640613867} a^{6} + \frac{3929232234245945}{29434370640613867} a^{5} - \frac{9612332799300}{29434370640613867} a^{4} + \frac{11199593672418048}{29434370640613867} a^{3} - \frac{2471742719820}{29434370640613867} a^{2} + \frac{5595519061880639}{29434370640613867} a - \frac{104861812356}{29434370640613867}$, $\frac{1}{29434370640613867} a^{31} - \frac{2548665}{29434370640613867} a^{23} - \frac{9879482312306227}{29434370640613867} a^{22} - \frac{134569512}{29434370640613867} a^{21} + \frac{12786063887478581}{29434370640613867} a^{20} - \frac{3211317900}{29434370640613867} a^{19} + \frac{4441816503929734}{29434370640613867} a^{18} - \frac{44827376400}{29434370640613867} a^{17} + \frac{7297849231671883}{29434370640613867} a^{16} - \frac{400084334370}{29434370640613867} a^{15} + \frac{1564198393937268}{29434370640613867} a^{14} - \frac{2358391865760}{29434370640613867} a^{13} + \frac{3020490268825338}{29434370640613867} a^{12} - \frac{9197728276464}{29434370640613867} a^{11} + \frac{786987202823463}{29434370640613867} a^{10} - \frac{23187550276800}{29434370640613867} a^{9} - \frac{13270379191793828}{29434370640613867} a^{8} - \frac{35868241834425}{29434370640613867} a^{7} - \frac{4543468306411700}{29434370640613867} a^{6} - \frac{30901869888120}{29434370640613867} a^{5} - \frac{4759417618223962}{29434370640613867} a^{4} - \frac{12297682914660}{29434370640613867} a^{3} + \frac{14339620972754397}{29434370640613867} a^{2} - \frac{1444762748016}{29434370640613867} a - \frac{11899983520636038}{29434370640613867}$, $\frac{1}{29434370640613867} a^{32} + \frac{9267253423391785}{29434370640613867} a^{23} + \frac{48934368}{29434370640613867} a^{22} + \frac{9364150823017394}{29434370640613867} a^{21} + \frac{2569054320}{29434370640613867} a^{20} - \frac{9912463508064275}{29434370640613867} a^{19} + \frac{59769835200}{29434370640613867} a^{18} - \frac{11515685040698895}{29434370640613867} a^{17} + \frac{800168668740}{29434370640613867} a^{16} - \frac{8335750665405848}{29434370640613867} a^{15} + \frac{6738262473600}{29434370640613867} a^{14} - \frac{7410411865348740}{29434370640613867} a^{13} + \frac{36790913105856}{29434370640613867} a^{12} - \frac{2055173374212863}{29434370640613867} a^{11} + \frac{129850281550080}{29434370640613867} a^{10} + \frac{431175659898826}{29434370640613867} a^{9} + \frac{286945934675400}{29434370640613867} a^{8} + \frac{2458708042266757}{29434370640613867} a^{7} + \frac{370822438657440}{29434370640613867} a^{6} + \frac{2242758730454495}{29434370640613867} a^{5} + \frac{245953658293200}{29434370640613867} a^{4} - 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\frac{2095406542658303}{29434370640613867} a^{17} + \frac{9765760378736575}{29434370640613867} a^{16} + \frac{12531314307783041}{29434370640613867} a^{15} - \frac{7461500048463134}{29434370640613867} a^{14} + \frac{3511560314418041}{29434370640613867} a^{13} + \frac{1068344579198138}{29434370640613867} a^{12} - \frac{5970416038098742}{29434370640613867} a^{11} - \frac{10041141126807400}{29434370640613867} a^{10} - \frac{14651506159249664}{29434370640613867} a^{9} - \frac{13021972337209496}{29434370640613867} a^{8} + \frac{6083532699309715}{29434370640613867} a^{7} + \frac{4573419368698133}{29434370640613867} a^{6} - \frac{8563472969322189}{29434370640613867} a^{5} + \frac{950169755088758}{29434370640613867} a^{4} - \frac{3696802356371276}{29434370640613867} a^{3} - \frac{4529181923270653}{29434370640613867} a^{2} + \frac{11730482679769818}{29434370640613867} a + \frac{8971412516344711}{29434370640613867}$, $\frac{1}{29434370640613867} a^{44} + \frac{2858888277256288}{29434370640613867} a^{23} + \frac{1358600419216404}{29434370640613867} a^{22} - \frac{2817958836800339}{29434370640613867} a^{21} - \frac{9843937712094270}{29434370640613867} a^{20} + \frac{4629798321055966}{29434370640613867} a^{19} - \frac{6312534039547609}{29434370640613867} a^{18} - \frac{3070976758998045}{29434370640613867} a^{17} - \frac{4954134546377185}{29434370640613867} a^{16} + \frac{335155957844288}{29434370640613867} a^{15} - \frac{9723076299887474}{29434370640613867} a^{14} - \frac{3223804657877642}{29434370640613867} a^{13} + \frac{7248041155694459}{29434370640613867} a^{12} - \frac{10688762505934359}{29434370640613867} a^{11} + \frac{5095894899254106}{29434370640613867} a^{10} - \frac{10199618619181343}{29434370640613867} a^{9} + \frac{2666826763651121}{29434370640613867} a^{8} + \frac{3766781933026035}{29434370640613867} a^{7} - \frac{8236693811824060}{29434370640613867} a^{6} + \frac{143532319416660}{29434370640613867} a^{5} + \frac{11230455279470169}{29434370640613867} a^{4} + \frac{4155253303531107}{29434370640613867} a^{3} + \frac{4579151427586301}{29434370640613867} a^{2} - \frac{13927013204139122}{29434370640613867} a + \frac{8287052634671398}{29434370640613867}$, $\frac{1}{29434370640613867} a^{45} + \frac{1798147613668770}{29434370640613867} a^{23} - \frac{2617320314956006}{29434370640613867} a^{22} - \frac{8949551935494883}{29434370640613867} a^{21} - \frac{3767273561154478}{29434370640613867} a^{20} - \frac{8915331382179866}{29434370640613867} a^{19} - \frac{6444501870183703}{29434370640613867} a^{18} + \frac{12350209239900341}{29434370640613867} a^{17} + \frac{10190666864001743}{29434370640613867} a^{16} + \frac{1013884227020333}{29434370640613867} a^{15} - \frac{5988276318089212}{29434370640613867} a^{14} - \frac{5896347077792493}{29434370640613867} a^{13} - \frac{1771303178748733}{29434370640613867} a^{12} - \frac{13659245502937038}{29434370640613867} a^{11} + \frac{12549553781056637}{29434370640613867} a^{10} + \frac{6274886502708520}{29434370640613867} a^{9} - \frac{12634368311564810}{29434370640613867} a^{8} + \frac{10031561974724805}{29434370640613867} a^{7} - \frac{13725816731493310}{29434370640613867} a^{6} + \frac{64340425405167}{29434370640613867} a^{5} - \frac{8965666892141569}{29434370640613867} a^{4} - \frac{574440043520623}{29434370640613867} a^{3} - \frac{12907651970512025}{29434370640613867} a^{2} - \frac{13822524850016904}{29434370640613867} a - \frac{3636822945342271}{29434370640613867}$

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:  $22$
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

\(\Q(\sqrt{-611}) \), \(\Q(\zeta_{47})^+\)

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