Properties

Label 46.0.732...779.1
Degree $46$
Signature $[0, 23]$
Discriminant $-7.330\times 10^{100}$
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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Normalized defining polynomial

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(y^46 - y^45 + 142*y^44 - 142*y^43 + 9448*y^42 - 9448*y^41 + 391417*y^40 - 391417*y^39 + 11317507*y^38 - 11317507*y^37 + 242638441*y^36 - 242638441*y^35 + 4000193125*y^34 - 4000193125*y^33 + 51909015346*y^32 - 51909015346*y^31 + 538367825590*y^30 - 538367825590*y^29 + 4506847593370*y^28 - 4506847593370*y^27 + 30634460334754*y^26 - 30634460334754*y^25 + 169585855368478*y^24 - 169585855368478*y^23 + 765091834084438*y^22 - 765091834084438*y^21 + 2810291100353278*y^20 - 2810291100353278*y^19 + 8388107281086478*y^18 - 8388107281086478*y^17 + 20310689367403693*y^16 - 20310689367403693*y^15 + 39925259896506208*y^14 - 39925259896506208*y^13 + 64155023491279903*y^12 - 64155023491279903*y^11 + 85878259817628733*y^10 - 85878259817628733*y^9 + 99353199644123308*y^8 - 99353199644123308*y^7 + 104743175574721138*y^6 - 104743175574721138*y^5 + 105987016174089868*y^4 - 105987016174089868*y^3 + 106122707875839184*y^2 - 106122707875839184*y + 106127132605244053, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{46} - x^{45} + 142 x^{44} - 142 x^{43} + 9448 x^{42} - 9448 x^{41} + 391417 x^{40} - 391417 x^{39} + 11317507 x^{38} - 11317507 x^{37} + 242638441 x^{36} + \cdots + 10\!\cdots\!53 \) Copy content Toggle raw display

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

Invariants

Degree:  $46$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 23]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-732\!\cdots\!779\) \(\medspace = -\,13^{23}\cdot 47^{45}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(155.85\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $13^{1/2}47^{45/46}\approx 155.85458413345435$
Ramified primes:   \(13\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-611}) \)
$\card{ \Gal(K/\Q) }$:  $46$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{29\!\cdots\!67}a^{24}+\frac{87\!\cdots\!49}{29\!\cdots\!67}a^{23}+\frac{72}{29\!\cdots\!67}a^{22}+\frac{14\!\cdots\!41}{29\!\cdots\!67}a^{21}+\frac{2268}{29\!\cdots\!67}a^{20}+\frac{81\!\cdots\!92}{29\!\cdots\!67}a^{19}+\frac{41040}{29\!\cdots\!67}a^{18}+\frac{31\!\cdots\!85}{29\!\cdots\!67}a^{17}+\frac{470934}{29\!\cdots\!67}a^{16}-\frac{22\!\cdots\!79}{29\!\cdots\!67}a^{15}+\frac{3569184}{29\!\cdots\!67}a^{14}+\frac{13\!\cdots\!14}{29\!\cdots\!67}a^{13}+\frac{18044208}{29\!\cdots\!67}a^{12}-\frac{17\!\cdots\!62}{29\!\cdots\!67}a^{11}+\frac{60046272}{29\!\cdots\!67}a^{10}+\frac{83\!\cdots\!27}{29\!\cdots\!67}a^{9}+\frac{126660105}{29\!\cdots\!67}a^{8}-\frac{84\!\cdots\!05}{29\!\cdots\!67}a^{7}+\frac{157621464}{29\!\cdots\!67}a^{6}-\frac{84\!\cdots\!05}{29\!\cdots\!67}a^{5}+\frac{101328084}{29\!\cdots\!67}a^{4}-\frac{16\!\cdots\!51}{29\!\cdots\!67}a^{3}+\frac{25509168}{29\!\cdots\!67}a^{2}-\frac{42\!\cdots\!57}{29\!\cdots\!67}a+\frac{1062882}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{25}+\frac{75}{29\!\cdots\!67}a^{23}+\frac{32\!\cdots\!20}{29\!\cdots\!67}a^{22}+\frac{2475}{29\!\cdots\!67}a^{21}+\frac{71\!\cdots\!51}{29\!\cdots\!67}a^{20}+\frac{47250}{29\!\cdots\!67}a^{19}+\frac{12\!\cdots\!18}{29\!\cdots\!67}a^{18}+\frac{577125}{29\!\cdots\!67}a^{17}-\frac{10\!\cdots\!95}{29\!\cdots\!67}a^{16}+\frac{4709340}{29\!\cdots\!67}a^{15}+\frac{14\!\cdots\!18}{29\!\cdots\!67}a^{14}+\frac{26025300}{29\!\cdots\!67}a^{13}-\frac{68\!\cdots\!40}{29\!\cdots\!67}a^{12}+\frac{96665400}{29\!\cdots\!67}a^{11}+\frac{13\!\cdots\!02}{29\!\cdots\!67}a^{10}+\frac{234555750}{29\!\cdots\!67}a^{9}-\frac{852737446735033}{29\!\cdots\!67}a^{8}+\frac{351833625}{29\!\cdots\!67}a^{7}+\frac{13\!\cdots\!84}{29\!\cdots\!67}a^{6}+\frac{295540245}{29\!\cdots\!67}a^{5}+\frac{10\!\cdots\!80}{29\!\cdots\!67}a^{4}+\frac{115145550}{29\!\cdots\!67}a^{3}+\frac{31\!\cdots\!04}{29\!\cdots\!67}a^{2}+\frac{13286025}{29\!\cdots\!67}a+\frac{11\!\cdots\!86}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{26}-\frac{43\!\cdots\!81}{29\!\cdots\!67}a^{23}-\frac{2925}{29\!\cdots\!67}a^{22}-\frac{13\!\cdots\!79}{29\!\cdots\!67}a^{21}-\frac{122850}{29\!\cdots\!67}a^{20}-\frac{60\!\cdots\!42}{29\!\cdots\!67}a^{19}-\frac{2500875}{29\!\cdots\!67}a^{18}-\frac{83\!\cdots\!34}{29\!\cdots\!67}a^{17}-\frac{30610710}{29\!\cdots\!67}a^{16}+\frac{35\!\cdots\!41}{29\!\cdots\!67}a^{15}-\frac{241663500}{29\!\cdots\!67}a^{14}+\frac{66\!\cdots\!22}{29\!\cdots\!67}a^{13}-\frac{1256650200}{29\!\cdots\!67}a^{12}-\frac{11\!\cdots\!83}{29\!\cdots\!67}a^{11}-\frac{4268914650}{29\!\cdots\!67}a^{10}-\frac{10\!\cdots\!51}{29\!\cdots\!67}a^{9}-\frac{9147674250}{29\!\cdots\!67}a^{8}+\frac{14\!\cdots\!85}{29\!\cdots\!67}a^{7}-\frac{11526069555}{29\!\cdots\!67}a^{6}-\frac{16\!\cdots\!19}{29\!\cdots\!67}a^{5}-\frac{7484460750}{29\!\cdots\!67}a^{4}+\frac{88\!\cdots\!61}{29\!\cdots\!67}a^{3}-\frac{1899901575}{29\!\cdots\!67}a^{2}+\frac{54\!\cdots\!24}{29\!\cdots\!67}a-\frac{79716150}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{27}-\frac{3159}{29\!\cdots\!67}a^{23}+\frac{32\!\cdots\!83}{29\!\cdots\!67}a^{22}-\frac{138996}{29\!\cdots\!67}a^{21}+\frac{83\!\cdots\!22}{29\!\cdots\!67}a^{20}-\frac{2985255}{29\!\cdots\!67}a^{19}+\frac{69\!\cdots\!34}{29\!\cdots\!67}a^{18}-\frac{38893608}{29\!\cdots\!67}a^{17}-\frac{93\!\cdots\!85}{29\!\cdots\!67}a^{16}-\frac{330595668}{29\!\cdots\!67}a^{15}+\frac{12\!\cdots\!77}{29\!\cdots\!67}a^{14}-\frac{1879175376}{29\!\cdots\!67}a^{13}-\frac{19\!\cdots\!70}{29\!\cdots\!67}a^{12}-\frac{7125206634}{29\!\cdots\!67}a^{11}+\frac{10\!\cdots\!93}{29\!\cdots\!67}a^{10}-\frac{17563534560}{29\!\cdots\!67}a^{9}-\frac{43\!\cdots\!91}{29\!\cdots\!67}a^{8}-\frac{26674618113}{29\!\cdots\!67}a^{7}-\frac{36\!\cdots\!75}{29\!\cdots\!67}a^{6}-\frac{22633009308}{29\!\cdots\!67}a^{5}-\frac{13\!\cdots\!80}{29\!\cdots\!67}a^{4}-\frac{8891539371}{29\!\cdots\!67}a^{3}-\frac{77\!\cdots\!61}{29\!\cdots\!67}a^{2}-\frac{1033121304}{29\!\cdots\!67}a-\frac{17\!\cdots\!98}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{28}-\frac{13\!\cdots\!72}{29\!\cdots\!67}a^{23}+\frac{88452}{29\!\cdots\!67}a^{22}+\frac{13\!\cdots\!40}{29\!\cdots\!67}a^{21}+\frac{4179357}{29\!\cdots\!67}a^{20}-\frac{12\!\cdots\!28}{29\!\cdots\!67}a^{19}+\frac{90751752}{29\!\cdots\!67}a^{18}-\frac{79\!\cdots\!48}{29\!\cdots\!67}a^{17}+\frac{1157084838}{29\!\cdots\!67}a^{16}+\frac{54086821103495}{29\!\cdots\!67}a^{15}+\frac{9395876880}{29\!\cdots\!67}a^{14}-\frac{22\!\cdots\!43}{29\!\cdots\!67}a^{13}+\frac{49876446438}{29\!\cdots\!67}a^{12}-\frac{11\!\cdots\!01}{29\!\cdots\!67}a^{11}+\frac{172122638688}{29\!\cdots\!67}a^{10}-\frac{11\!\cdots\!31}{29\!\cdots\!67}a^{9}+\frac{373444653582}{29\!\cdots\!67}a^{8}+\frac{12\!\cdots\!00}{29\!\cdots\!67}a^{7}+\frac{475293195468}{29\!\cdots\!67}a^{6}+\frac{25\!\cdots\!95}{29\!\cdots\!67}a^{5}+\frac{311203877985}{29\!\cdots\!67}a^{4}-\frac{413842854368211}{29\!\cdots\!67}a^{3}+\frac{79550340408}{29\!\cdots\!67}a^{2}+\frac{19\!\cdots\!24}{29\!\cdots\!67}a+\frac{3357644238}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{29}+\frac{98658}{29\!\cdots\!67}a^{23}+\frac{360746501103013}{29\!\cdots\!67}a^{22}+\frac{4883571}{29\!\cdots\!67}a^{21}-\frac{35\!\cdots\!07}{29\!\cdots\!67}a^{20}+\frac{111878172}{29\!\cdots\!67}a^{19}+\frac{20\!\cdots\!81}{29\!\cdots\!67}a^{18}+\frac{1518346620}{29\!\cdots\!67}a^{17}-\frac{12\!\cdots\!57}{29\!\cdots\!67}a^{16}+\frac{13274687592}{29\!\cdots\!67}a^{15}-\frac{11\!\cdots\!80}{29\!\cdots\!67}a^{14}+\frac{77028121422}{29\!\cdots\!67}a^{13}-\frac{108358777948730}{29\!\cdots\!67}a^{12}+\frac{296700912144}{29\!\cdots\!67}a^{11}+\frac{11\!\cdots\!25}{29\!\cdots\!67}a^{10}+\frac{740505637872}{29\!\cdots\!67}a^{9}+\frac{81\!\cdots\!17}{29\!\cdots\!67}a^{8}+\frac{1136002967190}{29\!\cdots\!67}a^{7}-\frac{98\!\cdots\!38}{29\!\cdots\!67}a^{6}+\frac{971913649707}{29\!\cdots\!67}a^{5}+\frac{21\!\cdots\!56}{29\!\cdots\!67}a^{4}+\frac{384493311972}{29\!\cdots\!67}a^{3}+\frac{10\!\cdots\!30}{29\!\cdots\!67}a^{2}+\frac{44940776724}{29\!\cdots\!67}a+\frac{28\!\cdots\!35}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{30}+\frac{36\!\cdots\!30}{29\!\cdots\!67}a^{23}-\frac{2219805}{29\!\cdots\!67}a^{22}+\frac{14\!\cdots\!97}{29\!\cdots\!67}a^{21}-\frac{111878172}{29\!\cdots\!67}a^{20}+\frac{11\!\cdots\!95}{29\!\cdots\!67}a^{19}-\frac{2530577700}{29\!\cdots\!67}a^{18}-\frac{78453107286076}{29\!\cdots\!67}a^{17}-\frac{33186718980}{29\!\cdots\!67}a^{16}-\frac{77\!\cdots\!03}{29\!\cdots\!67}a^{15}-\frac{275100433650}{29\!\cdots\!67}a^{14}-\frac{42\!\cdots\!58}{29\!\cdots\!67}a^{13}-\frac{1483504560720}{29\!\cdots\!67}a^{12}+\frac{62\!\cdots\!81}{29\!\cdots\!67}a^{11}-\frac{5183539465104}{29\!\cdots\!67}a^{10}-\frac{71\!\cdots\!93}{29\!\cdots\!67}a^{9}-\frac{11360029671900}{29\!\cdots\!67}a^{8}-\frac{80\!\cdots\!49}{29\!\cdots\!67}a^{7}-\frac{14578704745605}{29\!\cdots\!67}a^{6}+\frac{39\!\cdots\!45}{29\!\cdots\!67}a^{5}-\frac{9612332799300}{29\!\cdots\!67}a^{4}+\frac{11\!\cdots\!48}{29\!\cdots\!67}a^{3}-\frac{2471742719820}{29\!\cdots\!67}a^{2}+\frac{55\!\cdots\!39}{29\!\cdots\!67}a-\frac{104861812356}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{31}-\frac{2548665}{29\!\cdots\!67}a^{23}-\frac{98\!\cdots\!27}{29\!\cdots\!67}a^{22}-\frac{134569512}{29\!\cdots\!67}a^{21}+\frac{12\!\cdots\!81}{29\!\cdots\!67}a^{20}-\frac{3211317900}{29\!\cdots\!67}a^{19}+\frac{44\!\cdots\!34}{29\!\cdots\!67}a^{18}-\frac{44827376400}{29\!\cdots\!67}a^{17}+\frac{72\!\cdots\!83}{29\!\cdots\!67}a^{16}-\frac{400084334370}{29\!\cdots\!67}a^{15}+\frac{15\!\cdots\!68}{29\!\cdots\!67}a^{14}-\frac{2358391865760}{29\!\cdots\!67}a^{13}+\frac{30\!\cdots\!38}{29\!\cdots\!67}a^{12}-\frac{9197728276464}{29\!\cdots\!67}a^{11}+\frac{786987202823463}{29\!\cdots\!67}a^{10}-\frac{23187550276800}{29\!\cdots\!67}a^{9}-\frac{13\!\cdots\!28}{29\!\cdots\!67}a^{8}-\frac{35868241834425}{29\!\cdots\!67}a^{7}-\frac{45\!\cdots\!00}{29\!\cdots\!67}a^{6}-\frac{30901869888120}{29\!\cdots\!67}a^{5}-\frac{47\!\cdots\!62}{29\!\cdots\!67}a^{4}-\frac{12297682914660}{29\!\cdots\!67}a^{3}+\frac{14\!\cdots\!97}{29\!\cdots\!67}a^{2}-\frac{1444762748016}{29\!\cdots\!67}a-\frac{11\!\cdots\!38}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{32}+\frac{92\!\cdots\!85}{29\!\cdots\!67}a^{23}+\frac{48934368}{29\!\cdots\!67}a^{22}+\frac{93\!\cdots\!94}{29\!\cdots\!67}a^{21}+\frac{2569054320}{29\!\cdots\!67}a^{20}-\frac{99\!\cdots\!75}{29\!\cdots\!67}a^{19}+\frac{59769835200}{29\!\cdots\!67}a^{18}-\frac{11\!\cdots\!95}{29\!\cdots\!67}a^{17}+\frac{800168668740}{29\!\cdots\!67}a^{16}-\frac{83\!\cdots\!48}{29\!\cdots\!67}a^{15}+\frac{6738262473600}{29\!\cdots\!67}a^{14}-\frac{74\!\cdots\!40}{29\!\cdots\!67}a^{13}+\frac{36790913105856}{29\!\cdots\!67}a^{12}-\frac{20\!\cdots\!63}{29\!\cdots\!67}a^{11}+\frac{129850281550080}{29\!\cdots\!67}a^{10}+\frac{431175659898826}{29\!\cdots\!67}a^{9}+\frac{286945934675400}{29\!\cdots\!67}a^{8}+\frac{24\!\cdots\!57}{29\!\cdots\!67}a^{7}+\frac{370822438657440}{29\!\cdots\!67}a^{6}+\frac{22\!\cdots\!95}{29\!\cdots\!67}a^{5}+\frac{245953658293200}{29\!\cdots\!67}a^{4}-\frac{73\!\cdots\!18}{29\!\cdots\!67}a^{3}+\frac{63569560912704}{29\!\cdots\!67}a^{2}-\frac{54\!\cdots\!46}{29\!\cdots\!67}a+\frac{2708930152530}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{33}+\frac{57672648}{29\!\cdots\!67}a^{23}-\frac{10\!\cdots\!52}{29\!\cdots\!67}a^{22}+\frac{3171995640}{29\!\cdots\!67}a^{21}-\frac{11\!\cdots\!17}{29\!\cdots\!67}a^{20}+\frac{77858074800}{29\!\cdots\!67}a^{19}+\frac{11\!\cdots\!79}{29\!\cdots\!67}a^{18}+\frac{1109477565900}{29\!\cdots\!67}a^{17}-\frac{94\!\cdots\!81}{29\!\cdots\!67}a^{16}+\frac{10059263264160}{29\!\cdots\!67}a^{15}+\frac{10\!\cdots\!33}{29\!\cdots\!67}a^{14}+\frac{60037918639776}{29\!\cdots\!67}a^{13}+\frac{186978725074631}{29\!\cdots\!67}a^{12}+\frac{236513012823360}{29\!\cdots\!67}a^{11}-\frac{15\!\cdots\!12}{29\!\cdots\!67}a^{10}+\frac{601220053605600}{29\!\cdots\!67}a^{9}-\frac{99\!\cdots\!17}{29\!\cdots\!67}a^{8}+\frac{936515852731800}{29\!\cdots\!67}a^{7}+\frac{45\!\cdots\!12}{29\!\cdots\!67}a^{6}+\frac{811647072367560}{29\!\cdots\!67}a^{5}+\frac{433018265277043}{29\!\cdots\!67}a^{4}+\frac{324658828947024}{29\!\cdots\!67}a^{3}+\frac{10\!\cdots\!49}{29\!\cdots\!67}a^{2}+\frac{38312012157210}{29\!\cdots\!67}a+\frac{92\!\cdots\!11}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{34}+\frac{12\!\cdots\!74}{29\!\cdots\!67}a^{23}-\frac{980435016}{29\!\cdots\!67}a^{22}+\frac{13\!\cdots\!93}{29\!\cdots\!67}a^{21}-\frac{52943490864}{29\!\cdots\!67}a^{20}+\frac{48\!\cdots\!37}{29\!\cdots\!67}a^{19}-\frac{1257407908020}{29\!\cdots\!67}a^{18}+\frac{14\!\cdots\!69}{29\!\cdots\!67}a^{17}-\frac{17100747549072}{29\!\cdots\!67}a^{16}+\frac{62\!\cdots\!02}{29\!\cdots\!67}a^{15}-\frac{145806373839456}{29\!\cdots\!67}a^{14}+\frac{24\!\cdots\!81}{29\!\cdots\!67}a^{13}-\frac{804144243599424}{29\!\cdots\!67}a^{12}-\frac{13\!\cdots\!75}{29\!\cdots\!67}a^{11}-\frac{28\!\cdots\!56}{29\!\cdots\!67}a^{10}+\frac{29\!\cdots\!39}{29\!\cdots\!67}a^{9}-\frac{63\!\cdots\!40}{29\!\cdots\!67}a^{8}-\frac{13\!\cdots\!21}{29\!\cdots\!67}a^{7}-\frac{82\!\cdots\!12}{29\!\cdots\!67}a^{6}+\frac{11\!\cdots\!77}{29\!\cdots\!67}a^{5}-\frac{55\!\cdots\!08}{29\!\cdots\!67}a^{4}+\frac{87\!\cdots\!80}{29\!\cdots\!67}a^{3}-\frac{14\!\cdots\!54}{29\!\cdots\!67}a^{2}+\frac{13\!\cdots\!88}{29\!\cdots\!67}a-\frac{61299219451536}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{35}-\frac{1183283640}{29\!\cdots\!67}a^{23}-\frac{13\!\cdots\!92}{29\!\cdots\!67}a^{22}-\frac{66940045920}{29\!\cdots\!67}a^{21}+\frac{10\!\cdots\!19}{29\!\cdots\!67}a^{20}-\frac{1677304559700}{29\!\cdots\!67}a^{19}+\frac{13\!\cdots\!25}{29\!\cdots\!67}a^{18}-\frac{24280980292800}{29\!\cdots\!67}a^{17}+\frac{79\!\cdots\!28}{29\!\cdots\!67}a^{16}-\frac{222899399087904}{29\!\cdots\!67}a^{15}+\frac{10\!\cdots\!78}{29\!\cdots\!67}a^{14}-\frac{13\!\cdots\!60}{29\!\cdots\!67}a^{13}+\frac{608639831961754}{29\!\cdots\!67}a^{12}-\frac{53\!\cdots\!80}{29\!\cdots\!67}a^{11}-\frac{96\!\cdots\!17}{29\!\cdots\!67}a^{10}-\frac{13\!\cdots\!00}{29\!\cdots\!67}a^{9}+\frac{11\!\cdots\!75}{29\!\cdots\!67}a^{8}+\frac{80\!\cdots\!67}{29\!\cdots\!67}a^{7}+\frac{80\!\cdots\!22}{29\!\cdots\!67}a^{6}+\frac{10\!\cdots\!71}{29\!\cdots\!67}a^{5}-\frac{12\!\cdots\!77}{29\!\cdots\!67}a^{4}-\frac{74\!\cdots\!10}{29\!\cdots\!67}a^{3}+\frac{47\!\cdots\!58}{29\!\cdots\!67}a^{2}-\frac{887781798953280}{29\!\cdots\!67}a-\frac{27\!\cdots\!61}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{36}+\frac{81\!\cdots\!49}{29\!\cdots\!67}a^{23}+\frac{18256376160}{29\!\cdots\!67}a^{22}+\frac{12\!\cdots\!98}{29\!\cdots\!67}a^{21}+\frac{1006382735820}{29\!\cdots\!67}a^{20}-\frac{11\!\cdots\!76}{29\!\cdots\!67}a^{19}+\frac{24280980292800}{29\!\cdots\!67}a^{18}-\frac{50\!\cdots\!81}{29\!\cdots\!67}a^{17}+\frac{334349098631856}{29\!\cdots\!67}a^{16}-\frac{35\!\cdots\!33}{29\!\cdots\!67}a^{15}+\frac{28\!\cdots\!00}{29\!\cdots\!67}a^{14}-\frac{68\!\cdots\!22}{29\!\cdots\!67}a^{13}-\frac{13\!\cdots\!27}{29\!\cdots\!67}a^{12}-\frac{621036227766626}{29\!\cdots\!67}a^{11}-\frac{14\!\cdots\!54}{29\!\cdots\!67}a^{10}-\frac{24\!\cdots\!24}{29\!\cdots\!67}a^{9}+\frac{10\!\cdots\!32}{29\!\cdots\!67}a^{8}+\frac{914566084695604}{29\!\cdots\!67}a^{7}-\frac{87\!\cdots\!38}{29\!\cdots\!67}a^{6}+\frac{96\!\cdots\!72}{29\!\cdots\!67}a^{5}-\frac{53\!\cdots\!18}{29\!\cdots\!67}a^{4}+\frac{10\!\cdots\!78}{29\!\cdots\!67}a^{3}-\frac{137571275155627}{29\!\cdots\!67}a^{2}+\frac{33\!\cdots\!13}{29\!\cdots\!67}a+\frac{12\!\cdots\!80}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{37}+\frac{22516197264}{29\!\cdots\!67}a^{23}-\frac{13\!\cdots\!57}{29\!\cdots\!67}a^{22}+\frac{1300310391996}{29\!\cdots\!67}a^{21}+\frac{81\!\cdots\!68}{29\!\cdots\!67}a^{20}+\frac{33098809978080}{29\!\cdots\!67}a^{19}+\frac{27\!\cdots\!43}{29\!\cdots\!67}a^{18}+\frac{485133986250144}{29\!\cdots\!67}a^{17}-\frac{83\!\cdots\!12}{29\!\cdots\!67}a^{16}+\frac{44\!\cdots\!08}{29\!\cdots\!67}a^{15}+\frac{83\!\cdots\!43}{29\!\cdots\!67}a^{14}-\frac{20\!\cdots\!71}{29\!\cdots\!67}a^{13}-\frac{72\!\cdots\!65}{29\!\cdots\!67}a^{12}-\frac{83\!\cdots\!84}{29\!\cdots\!67}a^{11}-\frac{14\!\cdots\!51}{29\!\cdots\!67}a^{10}-\frac{12\!\cdots\!10}{29\!\cdots\!67}a^{9}+\frac{99\!\cdots\!94}{29\!\cdots\!67}a^{8}+\frac{21\!\cdots\!07}{29\!\cdots\!67}a^{7}+\frac{13\!\cdots\!26}{29\!\cdots\!67}a^{6}+\frac{55\!\cdots\!27}{29\!\cdots\!67}a^{5}-\frac{11\!\cdots\!97}{29\!\cdots\!67}a^{4}+\frac{94\!\cdots\!81}{29\!\cdots\!67}a^{3}+\frac{229441163793741}{29\!\cdots\!67}a^{2}-\frac{10\!\cdots\!63}{29\!\cdots\!67}a-\frac{129619593335753}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{38}-\frac{43\!\cdots\!61}{29\!\cdots\!67}a^{23}-\frac{320855811012}{29\!\cdots\!67}a^{22}-\frac{23\!\cdots\!49}{29\!\cdots\!67}a^{21}-\frac{17967925416672}{29\!\cdots\!67}a^{20}+\frac{12\!\cdots\!70}{29\!\cdots\!67}a^{19}-\frac{438930749464416}{29\!\cdots\!67}a^{18}+\frac{11\!\cdots\!28}{29\!\cdots\!67}a^{17}-\frac{61\!\cdots\!68}{29\!\cdots\!67}a^{16}-\frac{29\!\cdots\!06}{29\!\cdots\!67}a^{15}+\frac{58\!\cdots\!54}{29\!\cdots\!67}a^{14}+\frac{18\!\cdots\!93}{29\!\cdots\!67}a^{13}-\frac{25\!\cdots\!58}{29\!\cdots\!67}a^{12}-\frac{31\!\cdots\!54}{29\!\cdots\!67}a^{11}-\frac{10\!\cdots\!36}{29\!\cdots\!67}a^{10}-\frac{28\!\cdots\!31}{29\!\cdots\!67}a^{9}+\frac{53\!\cdots\!86}{29\!\cdots\!67}a^{8}+\frac{76\!\cdots\!68}{29\!\cdots\!67}a^{7}-\frac{11\!\cdots\!29}{29\!\cdots\!67}a^{6}-\frac{52\!\cdots\!55}{29\!\cdots\!67}a^{5}-\frac{56\!\cdots\!36}{29\!\cdots\!67}a^{4}-\frac{12\!\cdots\!68}{29\!\cdots\!67}a^{3}+\frac{34\!\cdots\!25}{29\!\cdots\!67}a^{2}-\frac{59\!\cdots\!77}{29\!\cdots\!67}a+\frac{55\!\cdots\!19}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{39}-\frac{403657310628}{29\!\cdots\!67}a^{23}+\frac{12\!\cdots\!73}{29\!\cdots\!67}a^{22}-\frac{23681228890176}{29\!\cdots\!67}a^{21}-\frac{61\!\cdots\!26}{29\!\cdots\!67}a^{20}-\frac{610329853669536}{29\!\cdots\!67}a^{19}+\frac{12\!\cdots\!36}{29\!\cdots\!67}a^{18}-\frac{90\!\cdots\!20}{29\!\cdots\!67}a^{17}-\frac{11\!\cdots\!35}{29\!\cdots\!67}a^{16}+\frac{38\!\cdots\!89}{29\!\cdots\!67}a^{15}-\frac{12\!\cdots\!49}{29\!\cdots\!67}a^{14}+\frac{12\!\cdots\!54}{29\!\cdots\!67}a^{13}+\frac{13\!\cdots\!81}{29\!\cdots\!67}a^{12}+\frac{87\!\cdots\!93}{29\!\cdots\!67}a^{11}+\frac{12\!\cdots\!85}{29\!\cdots\!67}a^{10}+\frac{241678619586813}{29\!\cdots\!67}a^{9}+\frac{416187423954956}{29\!\cdots\!67}a^{8}-\frac{14\!\cdots\!27}{29\!\cdots\!67}a^{7}-\frac{10\!\cdots\!01}{29\!\cdots\!67}a^{6}-\frac{89\!\cdots\!34}{29\!\cdots\!67}a^{5}+\frac{50\!\cdots\!36}{29\!\cdots\!67}a^{4}+\frac{19\!\cdots\!49}{29\!\cdots\!67}a^{3}+\frac{28\!\cdots\!80}{29\!\cdots\!67}a^{2}-\frac{80\!\cdots\!40}{29\!\cdots\!67}a-\frac{33\!\cdots\!31}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{40}+\frac{59\!\cdots\!31}{29\!\cdots\!67}a^{23}+\frac{5382097475040}{29\!\cdots\!67}a^{22}+\frac{10\!\cdots\!15}{29\!\cdots\!67}a^{21}+\frac{305164926834768}{29\!\cdots\!67}a^{20}-\frac{14\!\cdots\!40}{29\!\cdots\!67}a^{19}+\frac{75\!\cdots\!00}{29\!\cdots\!67}a^{18}-\frac{89\!\cdots\!66}{29\!\cdots\!67}a^{17}-\frac{12\!\cdots\!28}{29\!\cdots\!67}a^{16}-\frac{67\!\cdots\!33}{29\!\cdots\!67}a^{15}+\frac{11\!\cdots\!23}{29\!\cdots\!67}a^{14}-\frac{20\!\cdots\!41}{29\!\cdots\!67}a^{13}-\frac{72\!\cdots\!99}{29\!\cdots\!67}a^{12}+\frac{15\!\cdots\!73}{29\!\cdots\!67}a^{11}+\frac{13\!\cdots\!88}{29\!\cdots\!67}a^{10}-\frac{51\!\cdots\!38}{29\!\cdots\!67}a^{9}+\frac{14\!\cdots\!01}{29\!\cdots\!67}a^{8}-\frac{28\!\cdots\!30}{29\!\cdots\!67}a^{7}+\frac{84\!\cdots\!71}{29\!\cdots\!67}a^{6}+\frac{12\!\cdots\!07}{29\!\cdots\!67}a^{5}-\frac{99\!\cdots\!29}{29\!\cdots\!67}a^{4}+\frac{13\!\cdots\!59}{29\!\cdots\!67}a^{3}-\frac{13\!\cdots\!86}{29\!\cdots\!67}a^{2}-\frac{32\!\cdots\!96}{29\!\cdots\!67}a-\frac{12\!\cdots\!09}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{41}+\frac{6895812389895}{29\!\cdots\!67}a^{23}-\frac{61\!\cdots\!79}{29\!\cdots\!67}a^{22}+\frac{409611255959763}{29\!\cdots\!67}a^{21}-\frac{11\!\cdots\!95}{29\!\cdots\!67}a^{20}+\frac{10\!\cdots\!50}{29\!\cdots\!67}a^{19}+\frac{14\!\cdots\!29}{29\!\cdots\!67}a^{18}+\frac{12\!\cdots\!40}{29\!\cdots\!67}a^{17}+\frac{22\!\cdots\!10}{29\!\cdots\!67}a^{16}-\frac{23\!\cdots\!57}{29\!\cdots\!67}a^{15}-\frac{63\!\cdots\!44}{29\!\cdots\!67}a^{14}-\frac{12\!\cdots\!58}{29\!\cdots\!67}a^{13}-\frac{21\!\cdots\!29}{29\!\cdots\!67}a^{12}+\frac{60\!\cdots\!92}{29\!\cdots\!67}a^{11}-\frac{11\!\cdots\!68}{29\!\cdots\!67}a^{10}+\frac{20\!\cdots\!76}{29\!\cdots\!67}a^{9}+\frac{84\!\cdots\!88}{29\!\cdots\!67}a^{8}-\frac{14\!\cdots\!54}{29\!\cdots\!67}a^{7}+\frac{11\!\cdots\!78}{29\!\cdots\!67}a^{6}-\frac{30\!\cdots\!87}{29\!\cdots\!67}a^{5}-\frac{13\!\cdots\!67}{29\!\cdots\!67}a^{4}+\frac{36\!\cdots\!72}{29\!\cdots\!67}a^{3}-\frac{13\!\cdots\!28}{29\!\cdots\!67}a^{2}+\frac{31\!\cdots\!43}{29\!\cdots\!67}a+\frac{44\!\cdots\!14}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{42}-\frac{90\!\cdots\!24}{29\!\cdots\!67}a^{23}-\frac{86887236112677}{29\!\cdots\!67}a^{22}-\frac{60\!\cdots\!31}{29\!\cdots\!67}a^{21}-\frac{49\!\cdots\!10}{29\!\cdots\!67}a^{20}+\frac{53\!\cdots\!47}{29\!\cdots\!67}a^{19}-\frac{60\!\cdots\!57}{29\!\cdots\!67}a^{18}+\frac{94\!\cdots\!43}{29\!\cdots\!67}a^{17}-\frac{12\!\cdots\!17}{29\!\cdots\!67}a^{16}-\frac{33\!\cdots\!80}{29\!\cdots\!67}a^{15}+\frac{11\!\cdots\!41}{29\!\cdots\!67}a^{14}-\frac{10\!\cdots\!48}{29\!\cdots\!67}a^{13}-\frac{43\!\cdots\!59}{29\!\cdots\!67}a^{12}+\frac{10\!\cdots\!10}{29\!\cdots\!67}a^{11}-\frac{12\!\cdots\!75}{29\!\cdots\!67}a^{10}+\frac{13\!\cdots\!26}{29\!\cdots\!67}a^{9}-\frac{29\!\cdots\!71}{29\!\cdots\!67}a^{8}+\frac{11\!\cdots\!00}{29\!\cdots\!67}a^{7}-\frac{80\!\cdots\!58}{29\!\cdots\!67}a^{6}-\frac{23\!\cdots\!45}{29\!\cdots\!67}a^{5}+\frac{66\!\cdots\!05}{29\!\cdots\!67}a^{4}+\frac{90\!\cdots\!08}{29\!\cdots\!67}a^{3}-\frac{34\!\cdots\!25}{29\!\cdots\!67}a^{2}-\frac{13\!\cdots\!48}{29\!\cdots\!67}a-\frac{276575083524507}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{43}-\frac{113216701601367}{29\!\cdots\!67}a^{23}-\frac{20\!\cdots\!77}{29\!\cdots\!67}a^{22}-\frac{67\!\cdots\!20}{29\!\cdots\!67}a^{21}+\frac{13\!\cdots\!80}{29\!\cdots\!67}a^{20}-\frac{17\!\cdots\!23}{29\!\cdots\!67}a^{19}-\frac{11\!\cdots\!03}{29\!\cdots\!67}a^{18}-\frac{20\!\cdots\!03}{29\!\cdots\!67}a^{17}+\frac{97\!\cdots\!75}{29\!\cdots\!67}a^{16}+\frac{12\!\cdots\!41}{29\!\cdots\!67}a^{15}-\frac{74\!\cdots\!34}{29\!\cdots\!67}a^{14}+\frac{35\!\cdots\!41}{29\!\cdots\!67}a^{13}+\frac{10\!\cdots\!38}{29\!\cdots\!67}a^{12}-\frac{59\!\cdots\!42}{29\!\cdots\!67}a^{11}-\frac{10\!\cdots\!00}{29\!\cdots\!67}a^{10}-\frac{14\!\cdots\!64}{29\!\cdots\!67}a^{9}-\frac{13\!\cdots\!96}{29\!\cdots\!67}a^{8}+\frac{60\!\cdots\!15}{29\!\cdots\!67}a^{7}+\frac{45\!\cdots\!33}{29\!\cdots\!67}a^{6}-\frac{85\!\cdots\!89}{29\!\cdots\!67}a^{5}+\frac{950169755088758}{29\!\cdots\!67}a^{4}-\frac{36\!\cdots\!76}{29\!\cdots\!67}a^{3}-\frac{45\!\cdots\!53}{29\!\cdots\!67}a^{2}+\frac{11\!\cdots\!18}{29\!\cdots\!67}a+\frac{89\!\cdots\!11}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{44}+\frac{28\!\cdots\!88}{29\!\cdots\!67}a^{23}+\frac{13\!\cdots\!04}{29\!\cdots\!67}a^{22}-\frac{28\!\cdots\!39}{29\!\cdots\!67}a^{21}-\frac{98\!\cdots\!70}{29\!\cdots\!67}a^{20}+\frac{46\!\cdots\!66}{29\!\cdots\!67}a^{19}-\frac{63\!\cdots\!09}{29\!\cdots\!67}a^{18}-\frac{30\!\cdots\!45}{29\!\cdots\!67}a^{17}-\frac{49\!\cdots\!85}{29\!\cdots\!67}a^{16}+\frac{335155957844288}{29\!\cdots\!67}a^{15}-\frac{97\!\cdots\!74}{29\!\cdots\!67}a^{14}-\frac{32\!\cdots\!42}{29\!\cdots\!67}a^{13}+\frac{72\!\cdots\!59}{29\!\cdots\!67}a^{12}-\frac{10\!\cdots\!59}{29\!\cdots\!67}a^{11}+\frac{50\!\cdots\!06}{29\!\cdots\!67}a^{10}-\frac{10\!\cdots\!43}{29\!\cdots\!67}a^{9}+\frac{26\!\cdots\!21}{29\!\cdots\!67}a^{8}+\frac{37\!\cdots\!35}{29\!\cdots\!67}a^{7}-\frac{82\!\cdots\!60}{29\!\cdots\!67}a^{6}+\frac{143532319416660}{29\!\cdots\!67}a^{5}+\frac{11\!\cdots\!69}{29\!\cdots\!67}a^{4}+\frac{41\!\cdots\!07}{29\!\cdots\!67}a^{3}+\frac{45\!\cdots\!01}{29\!\cdots\!67}a^{2}-\frac{13\!\cdots\!22}{29\!\cdots\!67}a+\frac{82\!\cdots\!98}{29\!\cdots\!67}$, $\frac{1}{29\!\cdots\!67}a^{45}+\frac{17\!\cdots\!70}{29\!\cdots\!67}a^{23}-\frac{26\!\cdots\!06}{29\!\cdots\!67}a^{22}-\frac{89\!\cdots\!83}{29\!\cdots\!67}a^{21}-\frac{37\!\cdots\!78}{29\!\cdots\!67}a^{20}-\frac{89\!\cdots\!66}{29\!\cdots\!67}a^{19}-\frac{64\!\cdots\!03}{29\!\cdots\!67}a^{18}+\frac{12\!\cdots\!41}{29\!\cdots\!67}a^{17}+\frac{10\!\cdots\!43}{29\!\cdots\!67}a^{16}+\frac{10\!\cdots\!33}{29\!\cdots\!67}a^{15}-\frac{59\!\cdots\!12}{29\!\cdots\!67}a^{14}-\frac{58\!\cdots\!93}{29\!\cdots\!67}a^{13}-\frac{17\!\cdots\!33}{29\!\cdots\!67}a^{12}-\frac{13\!\cdots\!38}{29\!\cdots\!67}a^{11}+\frac{12\!\cdots\!37}{29\!\cdots\!67}a^{10}+\frac{62\!\cdots\!20}{29\!\cdots\!67}a^{9}-\frac{12\!\cdots\!10}{29\!\cdots\!67}a^{8}+\frac{10\!\cdots\!05}{29\!\cdots\!67}a^{7}-\frac{13\!\cdots\!10}{29\!\cdots\!67}a^{6}+\frac{64340425405167}{29\!\cdots\!67}a^{5}-\frac{89\!\cdots\!69}{29\!\cdots\!67}a^{4}-\frac{574440043520623}{29\!\cdots\!67}a^{3}-\frac{12\!\cdots\!25}{29\!\cdots\!67}a^{2}-\frac{13\!\cdots\!04}{29\!\cdots\!67}a-\frac{36\!\cdots\!71}{29\!\cdots\!67}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $22$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(13\) Copy content Toggle raw display Deg $46$$2$$23$$23$
\(47\) Copy content Toggle raw display Deg $46$$46$$1$$45$