LMFDB - Number field 40.0.184977318917298023500939014698578314068923764241165004603173974059593268961.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976)

gp: K = bnfinit(y^40 - y^39 + 6*y^38 - 17*y^37 + 17*y^36 - 102*y^35 + 73*y^34 - 73*y^33 + 438*y^32 + 2431*y^31 - 2431*y^30 - 14783*y^29 - 27726*y^28 - 119119*y^27 + 156703*y^26 - 53754*y^25 + 2550119*y^24 + 529177*y^23 + 6902634*y^22 - 17622319*y^21 - 42843857*y^20 - 105733914*y^19 + 248494824*y^18 + 114302232*y^17 + 3304954224*y^16 - 417991104*y^15 + 7311135168*y^14 - 33345696384*y^13 - 46569033216*y^12 - 148978579968*y^11 - 146993273856*y^10 + 881959643136*y^9 + 953430663168*y^8 - 953430663168*y^7 + 5720583979008*y^6 - 47958868426752*y^5 + 47958868426752*y^4 - 287753210560512*y^3 + 609359740010496*y^2 - 609359740010496*y + 3656158440062976, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976)

$ x^{40} - x^{39} + 6 x^{38} - 17 x^{37} + 17 x^{36} - 102 x^{35} + 73 x^{34} - 73 x^{33} + \cdots + 36\!\cdots\!76 $ x^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$40$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 20]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$184977318917298023500939014698578314068923764241165004603173974059593268961$ 184977318917298023500939014698578314068923764241165004603173974059593268961 $\medspace = 3^{20}\cdot 11^{36}\cdot 23^{20}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$71.89$	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:	$3^{1/2}11^{9/10}23^{1/2}\approx 71.89156900353078$
Ramified primes:	$3$, $11$, $23$ 3, 11, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$40$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$759=3\cdot 11\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{759}(1,·)$, $\chi_{759}(643,·)$, $\chi_{759}(392,·)$, $\chi_{759}(137,·)$, $\chi_{759}(139,·)$, $\chi_{759}(530,·)$, $\chi_{759}(277,·)$, $\chi_{759}(668,·)$, $\chi_{759}(413,·)$, $\chi_{759}(415,·)$, $\chi_{759}(160,·)$, $\chi_{759}(551,·)$, $\chi_{759}(553,·)$, $\chi_{759}(298,·)$, $\chi_{759}(47,·)$, $\chi_{759}(689,·)$, $\chi_{759}(691,·)$, $\chi_{759}(436,·)$, $\chi_{759}(185,·)$, $\chi_{759}(574,·)$, $\chi_{759}(323,·)$, $\chi_{759}(68,·)$, $\chi_{759}(70,·)$, $\chi_{759}(712,·)$, $\chi_{759}(461,·)$, $\chi_{759}(206,·)$, $\chi_{759}(208,·)$, $\chi_{759}(599,·)$, $\chi_{759}(344,·)$, $\chi_{759}(346,·)$, $\chi_{759}(91,·)$, $\chi_{759}(482,·)$, $\chi_{759}(229,·)$, $\chi_{759}(620,·)$, $\chi_{759}(622,·)$, $\chi_{759}(367,·)$, $\chi_{759}(116,·)$, $\chi_{759}(758,·)$, $\chi_{759}(505,·)$, $\chi_{759}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{6}a^{21}-\frac{1}{6}a^{20}+\frac{1}{6}a^{18}-\frac{1}{6}a^{17}+\frac{1}{6}a^{15}-\frac{1}{6}a^{14}+\frac{1}{6}a^{12}-\frac{1}{6}a^{11}+\frac{1}{6}a^{10}-\frac{1}{6}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{182124}a^{22}-\frac{1}{36}a^{21}-\frac{1}{3}a^{20}+\frac{1}{36}a^{19}+\frac{17}{36}a^{18}-\frac{1}{3}a^{17}-\frac{17}{36}a^{16}-\frac{1}{36}a^{15}-\frac{1}{3}a^{14}+\frac{1}{36}a^{13}+\frac{17}{36}a^{12}+\frac{78907}{182124}a^{11}+\frac{1}{3}a^{10}-\frac{13}{36}a^{9}-\frac{5}{36}a^{8}+\frac{1}{3}a^{7}+\frac{5}{36}a^{6}+\frac{13}{36}a^{5}+\frac{1}{3}a^{4}-\frac{13}{36}a^{3}-\frac{5}{36}a^{2}+\frac{1}{3}a+\frac{168}{5059}$, $\frac{1}{1092744}a^{23}-\frac{1}{1092744}a^{22}+\frac{1}{36}a^{21}+\frac{1}{216}a^{20}+\frac{107}{216}a^{19}-\frac{17}{36}a^{18}-\frac{17}{216}a^{17}-\frac{91}{216}a^{16}+\frac{1}{36}a^{15}+\frac{73}{216}a^{14}+\frac{35}{216}a^{13}+\frac{352093}{1092744}a^{12}+\frac{83813}{182124}a^{11}+\frac{95}{216}a^{10}+\frac{13}{216}a^{9}+\frac{5}{36}a^{8}-\frac{103}{216}a^{7}-\frac{5}{216}a^{6}-\frac{13}{36}a^{5}+\frac{23}{216}a^{4}+\frac{85}{216}a^{3}+\frac{5}{36}a^{2}-\frac{5003}{10118}a-\frac{28}{5059}$, $\frac{1}{6556464}a^{24}-\frac{1}{6556464}a^{23}+\frac{1}{1092744}a^{22}-\frac{107}{1296}a^{21}+\frac{107}{1296}a^{20}-\frac{107}{216}a^{19}+\frac{307}{1296}a^{18}-\frac{307}{1296}a^{17}+\frac{91}{216}a^{16}-\frac{251}{1296}a^{15}+\frac{251}{1296}a^{14}+\frac{3083953}{6556464}a^{13}+\frac{539123}{1092744}a^{12}-\frac{2172415}{6556464}a^{11}-\frac{635}{1296}a^{10}-\frac{13}{216}a^{9}-\frac{211}{1296}a^{8}+\frac{211}{1296}a^{7}+\frac{5}{216}a^{6}-\frac{85}{1296}a^{5}+\frac{85}{1296}a^{4}-\frac{85}{216}a^{3}-\frac{1677}{10118}a^{2}+\frac{1677}{10118}a+\frac{28}{5059}$, $\frac{1}{39338784}a^{25}-\frac{1}{39338784}a^{24}+\frac{1}{6556464}a^{23}-\frac{17}{39338784}a^{22}+\frac{539}{7776}a^{21}-\frac{107}{1296}a^{20}+\frac{2467}{7776}a^{19}+\frac{2717}{7776}a^{18}+\frac{307}{1296}a^{17}-\frac{3275}{7776}a^{16}+\frac{683}{7776}a^{15}-\frac{3472511}{39338784}a^{14}-\frac{3103357}{6556464}a^{13}-\frac{13099855}{39338784}a^{12}+\frac{13137439}{39338784}a^{11}+\frac{635}{1296}a^{10}+\frac{1517}{7776}a^{9}+\frac{1075}{7776}a^{8}-\frac{211}{1296}a^{7}+\frac{347}{7776}a^{6}+\frac{2245}{7776}a^{5}-\frac{85}{1296}a^{4}+\frac{65795}{182124}a^{3}-\frac{5087}{182124}a^{2}-\frac{1677}{10118}a+\frac{1111}{5059}$, $\frac{1}{236032704}a^{26}-\frac{1}{236032704}a^{25}+\frac{1}{39338784}a^{24}-\frac{17}{236032704}a^{23}+\frac{17}{236032704}a^{22}-\frac{539}{7776}a^{21}+\frac{10243}{46656}a^{20}+\frac{20861}{46656}a^{19}-\frac{2717}{7776}a^{18}-\frac{11051}{46656}a^{17}-\frac{4501}{46656}a^{16}+\frac{101430913}{236032704}a^{15}+\frac{3453107}{39338784}a^{14}+\frac{39351857}{236032704}a^{13}+\frac{39363295}{236032704}a^{12}-\frac{12989695}{39338784}a^{11}-\frac{21811}{46656}a^{10}-\frac{9293}{46656}a^{9}-\frac{1075}{7776}a^{8}+\frac{347}{46656}a^{7}+\frac{15205}{46656}a^{6}-\frac{2245}{7776}a^{5}-\frac{116329}{1092744}a^{4}+\frac{480577}{1092744}a^{3}+\frac{5087}{182124}a^{2}+\frac{3085}{15177}a+\frac{658}{5059}$, $\frac{1}{1416196224}a^{27}-\frac{1}{1416196224}a^{26}+\frac{1}{236032704}a^{25}-\frac{17}{1416196224}a^{24}+\frac{17}{1416196224}a^{23}-\frac{17}{236032704}a^{22}-\frac{20861}{279936}a^{21}+\frac{20861}{279936}a^{20}-\frac{20861}{46656}a^{19}-\frac{88811}{279936}a^{18}+\frac{88811}{279936}a^{17}-\frac{449312063}{1416196224}a^{16}-\frac{22772749}{236032704}a^{15}+\frac{39351857}{1416196224}a^{14}-\frac{39314273}{1416196224}a^{13}+\frac{39462017}{236032704}a^{12}+\frac{870503}{1416196224}a^{11}-\frac{55949}{279936}a^{10}+\frac{9293}{46656}a^{9}-\frac{61861}{279936}a^{8}+\frac{61861}{279936}a^{7}-\frac{15205}{46656}a^{6}+\frac{2797655}{6556464}a^{5}-\frac{2797655}{6556464}a^{4}-\frac{480577}{1092744}a^{3}+\frac{14519}{30354}a^{2}-\frac{14519}{30354}a-\frac{658}{5059}$, $\frac{1}{8497177344}a^{28}-\frac{1}{8497177344}a^{27}+\frac{1}{1416196224}a^{26}-\frac{17}{8497177344}a^{25}+\frac{17}{8497177344}a^{24}-\frac{17}{1416196224}a^{23}+\frac{73}{8497177344}a^{22}+\frac{20861}{1679616}a^{21}+\frac{25795}{279936}a^{20}+\frac{750997}{1679616}a^{19}+\frac{648683}{1679616}a^{18}-\frac{3281704511}{8497177344}a^{17}+\frac{449292659}{1416196224}a^{16}+\frac{2871744305}{8497177344}a^{15}-\frac{2871706721}{8497177344}a^{14}+\frac{39462017}{1416196224}a^{13}-\frac{1415325721}{8497177344}a^{12}+\frac{1418405017}{8497177344}a^{11}+\frac{9293}{279936}a^{10}-\frac{341797}{1679616}a^{9}+\frac{621733}{1679616}a^{8}-\frac{108517}{279936}a^{7}+\frac{9354119}{39338784}a^{6}-\frac{2797655}{39338784}a^{5}+\frac{1704911}{6556464}a^{4}-\frac{15835}{182124}a^{3}+\frac{46189}{182124}a^{2}+\frac{4730}{15177}a+\frac{591}{5059}$, $\frac{1}{50983064064}a^{29}-\frac{1}{50983064064}a^{28}+\frac{1}{8497177344}a^{27}-\frac{17}{50983064064}a^{26}+\frac{17}{50983064064}a^{25}-\frac{17}{8497177344}a^{24}+\frac{73}{50983064064}a^{23}-\frac{73}{50983064064}a^{22}-\frac{20861}{1679616}a^{21}-\frac{928619}{10077696}a^{20}-\frac{750997}{10077696}a^{19}+\frac{12296453953}{50983064064}a^{18}+\frac{3281685107}{8497177344}a^{17}-\frac{21203591503}{50983064064}a^{16}-\frac{21282257633}{50983064064}a^{15}+\frac{2871854465}{8497177344}a^{14}+\frac{8498047847}{50983064064}a^{13}+\frac{2208793}{50983064064}a^{12}-\frac{1416725401}{8497177344}a^{11}-\frac{341797}{10077696}a^{10}+\frac{2021413}{10077696}a^{9}-\frac{621733}{1679616}a^{8}+\frac{88031687}{236032704}a^{7}-\frac{48692903}{236032704}a^{6}+\frac{2797655}{39338784}a^{5}-\frac{15835}{1092744}a^{4}+\frac{197959}{1092744}a^{3}-\frac{46189}{182124}a^{2}-\frac{9527}{30354}a+\frac{2431}{5059}$, $\frac{1}{1529491921920}a^{30}+\frac{1}{305898384384}a^{29}+\frac{19}{1529491921920}a^{27}-\frac{17}{305898384384}a^{26}-\frac{539}{1529491921920}a^{24}+\frac{73}{305898384384}a^{23}+\frac{1}{302330880}a^{21}+\frac{10828693}{60466176}a^{20}+\frac{6397315547}{305898384384}a^{19}-\frac{539}{7080981120}a^{18}+\frac{1816723}{60466176}a^{17}+\frac{121548995393}{305898384384}a^{16}+\frac{19}{32782320}a^{15}-\frac{11701115}{60466176}a^{14}-\frac{83270851609}{305898384384}a^{13}+\frac{1}{151770}a^{12}-\frac{12094685}{60466176}a^{11}-\frac{12099109}{60466176}a^{10}+\frac{1}{5}a^{9}-\frac{1703594945}{8497177344}a^{8}+\frac{284725607}{1416196224}a^{7}-\frac{1}{5}a^{6}+\frac{8189737}{39338784}a^{5}-\frac{1290703}{6556464}a^{4}+\frac{1}{5}a^{3}-\frac{42509}{182124}a^{2}-\frac{2431}{30354}a-\frac{1}{5}$, $\frac{1}{91\!\cdots\!20}a^{31}-\frac{125315581}{91\!\cdots\!20}a^{30}+\frac{88014187}{10\!\cdots\!88}a^{29}-\frac{8547854741}{91\!\cdots\!20}a^{28}+\frac{48906132341}{91\!\cdots\!20}a^{27}-\frac{1496241179}{10\!\cdots\!88}a^{26}+\frac{145313530381}{91\!\cdots\!20}a^{25}-\frac{804336084301}{91\!\cdots\!20}a^{24}+\frac{6425035651}{10\!\cdots\!88}a^{23}-\frac{623993392421}{91\!\cdots\!20}a^{22}-\frac{10\!\cdots\!01}{18\!\cdots\!80}a^{21}+\frac{52\!\cdots\!81}{18\!\cdots\!84}a^{20}+\frac{19\!\cdots\!57}{50\!\cdots\!40}a^{19}-\frac{27\!\cdots\!11}{91\!\cdots\!20}a^{18}-\frac{80\!\cdots\!45}{18\!\cdots\!84}a^{17}+\frac{13\!\cdots\!23}{50\!\cdots\!40}a^{16}-\frac{11\!\cdots\!29}{91\!\cdots\!20}a^{15}+\frac{10\!\cdots\!89}{18\!\cdots\!84}a^{14}+\frac{23\!\cdots\!37}{50\!\cdots\!40}a^{13}-\frac{59\!\cdots\!91}{91\!\cdots\!20}a^{12}+\frac{86\!\cdots\!95}{18\!\cdots\!84}a^{11}-\frac{33\!\cdots\!23}{10\!\cdots\!60}a^{10}+\frac{261321235252883}{94\!\cdots\!60}a^{9}+\frac{23\!\cdots\!39}{70\!\cdots\!52}a^{8}-\frac{77\!\cdots\!23}{23\!\cdots\!40}a^{7}-\frac{13071201798763}{43\!\cdots\!60}a^{6}+\frac{114539454959701}{327653456425272}a^{5}-\frac{354998423490617}{10\!\cdots\!40}a^{4}-\frac{181068540157}{20225522001560}a^{3}+\frac{199236135670}{505638050039}a^{2}-\frac{1111424224769}{15169141501170}a-\frac{509519377516}{2528190250195}$, $\frac{1}{55\!\cdots\!20}a^{32}-\frac{1}{55\!\cdots\!20}a^{31}-\frac{262300697}{91\!\cdots\!20}a^{30}+\frac{31110785299}{55\!\cdots\!20}a^{29}-\frac{38979806239}{55\!\cdots\!20}a^{28}+\frac{33996092977}{91\!\cdots\!20}a^{27}-\frac{528883350299}{55\!\cdots\!20}a^{26}+\frac{662656706279}{55\!\cdots\!20}a^{25}-\frac{521276630057}{91\!\cdots\!20}a^{24}+\frac{2271087330499}{55\!\cdots\!20}a^{23}-\frac{2845525859119}{55\!\cdots\!20}a^{22}-\frac{45\!\cdots\!03}{55\!\cdots\!20}a^{21}+\frac{18\!\cdots\!41}{91\!\cdots\!20}a^{20}+\frac{91\!\cdots\!29}{55\!\cdots\!20}a^{19}-\frac{24\!\cdots\!77}{55\!\cdots\!20}a^{18}+\frac{32\!\cdots\!99}{91\!\cdots\!20}a^{17}+\frac{23\!\cdots\!11}{55\!\cdots\!20}a^{16}+\frac{23\!\cdots\!57}{55\!\cdots\!20}a^{15}-\frac{40\!\cdots\!59}{91\!\cdots\!20}a^{14}-\frac{12\!\cdots\!91}{55\!\cdots\!20}a^{13}+\frac{50\!\cdots\!83}{55\!\cdots\!20}a^{12}-\frac{53\!\cdots\!81}{91\!\cdots\!20}a^{11}-\frac{27\!\cdots\!83}{62\!\cdots\!40}a^{10}-\frac{10\!\cdots\!73}{25\!\cdots\!20}a^{9}+\frac{24\!\cdots\!09}{10\!\cdots\!80}a^{8}-\frac{674291542989217}{29\!\cdots\!40}a^{7}+\frac{46\!\cdots\!73}{11\!\cdots\!20}a^{6}+\frac{21\!\cdots\!71}{49\!\cdots\!80}a^{5}-\frac{51927339873127}{121353132009360}a^{4}-\frac{222151315454873}{546089094042120}a^{3}+\frac{13141192097653}{45507424503510}a^{2}+\frac{2141258763287}{7584570750585}a+\frac{17338242028}{2528190250195}$, $\frac{1}{33\!\cdots\!20}a^{33}-\frac{1}{33\!\cdots\!20}a^{32}+\frac{1}{55\!\cdots\!20}a^{31}+\frac{9942566167}{33\!\cdots\!20}a^{30}+\frac{323332495001}{33\!\cdots\!20}a^{29}-\frac{45603277361}{55\!\cdots\!20}a^{28}+\frac{1830626741953}{33\!\cdots\!20}a^{27}-\frac{5496652414801}{33\!\cdots\!20}a^{26}+\frac{775255714921}{55\!\cdots\!20}a^{25}-\frac{33268248905273}{33\!\cdots\!20}a^{24}+\frac{23603272131401}{33\!\cdots\!20}a^{23}-\frac{70\!\cdots\!51}{33\!\cdots\!20}a^{22}-\frac{43\!\cdots\!37}{55\!\cdots\!20}a^{21}+\frac{12\!\cdots\!49}{33\!\cdots\!20}a^{20}+\frac{64\!\cdots\!91}{33\!\cdots\!20}a^{19}-\frac{21\!\cdots\!83}{55\!\cdots\!20}a^{18}-\frac{12\!\cdots\!89}{33\!\cdots\!20}a^{17}-\frac{15\!\cdots\!91}{33\!\cdots\!20}a^{16}-\frac{24\!\cdots\!77}{55\!\cdots\!20}a^{15}+\frac{14\!\cdots\!29}{33\!\cdots\!20}a^{14}+\frac{12\!\cdots\!91}{33\!\cdots\!20}a^{13}+\frac{35\!\cdots\!97}{55\!\cdots\!20}a^{12}+\frac{72\!\cdots\!33}{76\!\cdots\!60}a^{11}-\frac{15\!\cdots\!69}{38\!\cdots\!80}a^{10}+\frac{59\!\cdots\!23}{25\!\cdots\!20}a^{9}-\frac{99\!\cdots\!01}{42\!\cdots\!20}a^{8}+\frac{71\!\cdots\!49}{17\!\cdots\!80}a^{7}+\frac{50\!\cdots\!57}{11\!\cdots\!20}a^{6}-\frac{81\!\cdots\!19}{19\!\cdots\!20}a^{5}-\frac{319948188593029}{819133641063180}a^{4}+\frac{123073740639023}{546089094042120}a^{3}-\frac{18290040903871}{91014849007020}a^{2}-\frac{3329385139532}{7584570750585}a+\frac{1219021251933}{2528190250195}$, $\frac{1}{19\!\cdots\!20}a^{34}-\frac{1}{19\!\cdots\!20}a^{33}+\frac{1}{33\!\cdots\!20}a^{32}-\frac{17}{19\!\cdots\!20}a^{31}+\frac{905824429}{39\!\cdots\!44}a^{30}-\frac{161819130041}{33\!\cdots\!20}a^{29}+\frac{993560390857}{19\!\cdots\!20}a^{28}-\frac{1175061804869}{39\!\cdots\!44}a^{27}+\frac{2750925210481}{33\!\cdots\!20}a^{26}-\frac{16890526640897}{19\!\cdots\!20}a^{25}+\frac{19780392606109}{39\!\cdots\!44}a^{24}-\frac{70\!\cdots\!91}{19\!\cdots\!20}a^{23}+\frac{11\!\cdots\!67}{33\!\cdots\!20}a^{22}+\frac{26\!\cdots\!81}{39\!\cdots\!44}a^{21}-\frac{70\!\cdots\!29}{19\!\cdots\!20}a^{20}-\frac{13\!\cdots\!87}{33\!\cdots\!20}a^{19}-\frac{14\!\cdots\!45}{39\!\cdots\!44}a^{18}+\frac{96\!\cdots\!49}{19\!\cdots\!20}a^{17}+\frac{14\!\cdots\!07}{33\!\cdots\!20}a^{16}+\frac{54\!\cdots\!21}{39\!\cdots\!44}a^{15}+\frac{60\!\cdots\!11}{19\!\cdots\!20}a^{14}+\frac{93\!\cdots\!33}{33\!\cdots\!20}a^{13}+\frac{20\!\cdots\!31}{18\!\cdots\!84}a^{12}-\frac{21\!\cdots\!01}{91\!\cdots\!20}a^{11}-\frac{43\!\cdots\!11}{19\!\cdots\!40}a^{10}-\frac{26\!\cdots\!51}{25\!\cdots\!72}a^{9}-\frac{68\!\cdots\!27}{21\!\cdots\!60}a^{8}-\frac{29\!\cdots\!29}{88\!\cdots\!40}a^{7}-\frac{46539042521141}{11\!\cdots\!92}a^{6}+\frac{35\!\cdots\!67}{98\!\cdots\!60}a^{5}-\frac{190436717692637}{819133641063180}a^{4}-\frac{2750596184599}{27304454702106}a^{3}-\frac{1442738591396}{22753712251755}a^{2}-\frac{137178932798}{7584570750585}a-\frac{821763089861}{2528190250195}$, $\frac{1}{11\!\cdots\!20}a^{35}-\frac{1}{11\!\cdots\!20}a^{34}+\frac{1}{19\!\cdots\!20}a^{33}-\frac{17}{11\!\cdots\!20}a^{32}+\frac{17}{11\!\cdots\!20}a^{31}+\frac{2660043871}{19\!\cdots\!20}a^{30}-\frac{3929121828503}{11\!\cdots\!20}a^{29}+\frac{4008923145143}{11\!\cdots\!20}a^{28}-\frac{3958382311271}{19\!\cdots\!20}a^{27}+\frac{66795071088223}{11\!\cdots\!20}a^{26}-\frac{68151693471103}{11\!\cdots\!20}a^{25}-\frac{69\!\cdots\!39}{11\!\cdots\!20}a^{24}+\frac{11\!\cdots\!87}{19\!\cdots\!20}a^{23}-\frac{41\!\cdots\!27}{11\!\cdots\!20}a^{22}+\frac{26\!\cdots\!19}{11\!\cdots\!20}a^{21}+\frac{47\!\cdots\!13}{19\!\cdots\!20}a^{20}-\frac{11\!\cdots\!33}{11\!\cdots\!20}a^{19}+\frac{56\!\cdots\!61}{11\!\cdots\!20}a^{18}-\frac{38\!\cdots\!53}{19\!\cdots\!20}a^{17}-\frac{58\!\cdots\!87}{11\!\cdots\!20}a^{16}+\frac{11\!\cdots\!19}{11\!\cdots\!20}a^{15}+\frac{81\!\cdots\!73}{19\!\cdots\!20}a^{14}-\frac{26\!\cdots\!73}{55\!\cdots\!20}a^{13}-\frac{15\!\cdots\!39}{55\!\cdots\!20}a^{12}-\frac{42\!\cdots\!83}{91\!\cdots\!20}a^{11}+\frac{43\!\cdots\!73}{15\!\cdots\!20}a^{10}+\frac{77\!\cdots\!59}{25\!\cdots\!20}a^{9}-\frac{46\!\cdots\!69}{23\!\cdots\!40}a^{8}+\frac{14\!\cdots\!27}{70\!\cdots\!20}a^{7}-\frac{22\!\cdots\!79}{11\!\cdots\!20}a^{6}+\frac{412767828890339}{10\!\cdots\!40}a^{5}-\frac{12\!\cdots\!97}{32\!\cdots\!20}a^{4}+\frac{168629737877509}{546089094042120}a^{3}+\frac{726373696042}{7584570750585}a^{2}+\frac{422973765376}{2528190250195}a+\frac{1027499163912}{2528190250195}$, $\frac{1}{71\!\cdots\!20}a^{36}-\frac{1}{71\!\cdots\!20}a^{35}+\frac{1}{11\!\cdots\!20}a^{34}-\frac{17}{71\!\cdots\!20}a^{33}+\frac{17}{71\!\cdots\!20}a^{32}-\frac{17}{11\!\cdots\!20}a^{31}-\frac{1939306515191}{71\!\cdots\!20}a^{30}-\frac{21472376059657}{71\!\cdots\!20}a^{29}+\frac{1962640580617}{11\!\cdots\!20}a^{28}-\frac{107501884687169}{71\!\cdots\!20}a^{27}+\frac{365030393010497}{71\!\cdots\!20}a^{26}-\frac{70\!\cdots\!67}{71\!\cdots\!20}a^{25}+\frac{12\!\cdots\!99}{11\!\cdots\!20}a^{24}-\frac{42\!\cdots\!27}{71\!\cdots\!20}a^{23}+\frac{11\!\cdots\!07}{71\!\cdots\!20}a^{22}+\frac{51\!\cdots\!61}{11\!\cdots\!20}a^{21}+\frac{69\!\cdots\!67}{71\!\cdots\!20}a^{20}-\frac{29\!\cdots\!47}{71\!\cdots\!20}a^{19}-\frac{56\!\cdots\!81}{11\!\cdots\!20}a^{18}+\frac{17\!\cdots\!53}{71\!\cdots\!20}a^{17}+\frac{24\!\cdots\!67}{71\!\cdots\!20}a^{16}-\frac{21\!\cdots\!79}{11\!\cdots\!20}a^{15}+\frac{10\!\cdots\!27}{33\!\cdots\!20}a^{14}-\frac{83\!\cdots\!67}{33\!\cdots\!20}a^{13}-\frac{11\!\cdots\!61}{55\!\cdots\!20}a^{12}-\frac{19\!\cdots\!57}{42\!\cdots\!20}a^{11}+\frac{14\!\cdots\!61}{12\!\cdots\!60}a^{10}+\frac{13\!\cdots\!97}{31\!\cdots\!40}a^{9}-\frac{27\!\cdots\!57}{10\!\cdots\!80}a^{8}+\frac{23\!\cdots\!19}{58\!\cdots\!60}a^{7}+\frac{14\!\cdots\!31}{29\!\cdots\!80}a^{6}-\frac{32\!\cdots\!21}{98\!\cdots\!60}a^{5}-\frac{119392997822933}{546089094042120}a^{4}-\frac{47963064145007}{136522273510530}a^{3}-\frac{10831907219386}{22753712251755}a^{2}-\frac{396876624241}{15169141501170}a-\frac{949039810578}{2528190250195}$, $\frac{1}{42\!\cdots\!20}a^{37}-\frac{1}{42\!\cdots\!20}a^{36}+\frac{1}{71\!\cdots\!20}a^{35}-\frac{17}{42\!\cdots\!20}a^{34}+\frac{17}{42\!\cdots\!20}a^{33}-\frac{17}{71\!\cdots\!20}a^{32}+\frac{73}{42\!\cdots\!20}a^{31}-\frac{11578354513993}{42\!\cdots\!20}a^{30}-\frac{19446888727223}{71\!\cdots\!20}a^{29}+\frac{58789559796607}{42\!\cdots\!20}a^{28}-\frac{572726094531967}{42\!\cdots\!20}a^{27}-\frac{68\!\cdots\!87}{42\!\cdots\!20}a^{26}+\frac{11\!\cdots\!63}{71\!\cdots\!20}a^{25}-\frac{40\!\cdots\!63}{42\!\cdots\!20}a^{24}+\frac{11\!\cdots\!87}{42\!\cdots\!20}a^{23}-\frac{19\!\cdots\!23}{71\!\cdots\!20}a^{22}-\frac{23\!\cdots\!77}{42\!\cdots\!20}a^{21}-\frac{40\!\cdots\!67}{42\!\cdots\!20}a^{20}+\frac{25\!\cdots\!03}{71\!\cdots\!20}a^{19}+\frac{32\!\cdots\!17}{42\!\cdots\!20}a^{18}+\frac{10\!\cdots\!67}{42\!\cdots\!20}a^{17}+\frac{26\!\cdots\!17}{71\!\cdots\!20}a^{16}+\frac{69\!\cdots\!83}{19\!\cdots\!20}a^{15}+\frac{65\!\cdots\!33}{19\!\cdots\!20}a^{14}+\frac{25\!\cdots\!03}{33\!\cdots\!20}a^{13}-\frac{10\!\cdots\!43}{91\!\cdots\!20}a^{12}-\frac{22\!\cdots\!33}{91\!\cdots\!20}a^{11}+\frac{24\!\cdots\!39}{50\!\cdots\!40}a^{10}+\frac{25\!\cdots\!33}{70\!\cdots\!20}a^{9}-\frac{70\!\cdots\!39}{14\!\cdots\!40}a^{8}-\frac{75\!\cdots\!39}{23\!\cdots\!40}a^{7}+\frac{29\!\cdots\!37}{19\!\cdots\!20}a^{6}+\frac{16\!\cdots\!89}{65\!\cdots\!40}a^{5}-\frac{540377386372429}{10\!\cdots\!40}a^{4}+\frac{29744028342007}{91014849007020}a^{3}+\frac{6664301308753}{30338283002340}a^{2}+\frac{2234592350983}{7584570750585}a-\frac{873195350961}{2528190250195}$, $\frac{1}{25\!\cdots\!20}a^{38}-\frac{1}{25\!\cdots\!20}a^{37}+\frac{1}{42\!\cdots\!20}a^{36}-\frac{17}{25\!\cdots\!20}a^{35}+\frac{17}{25\!\cdots\!20}a^{34}-\frac{17}{42\!\cdots\!20}a^{33}+\frac{73}{25\!\cdots\!20}a^{32}-\frac{73}{25\!\cdots\!20}a^{31}-\frac{2061090105911}{42\!\cdots\!20}a^{30}+\frac{417585930048127}{25\!\cdots\!20}a^{29}-\frac{479418633227647}{25\!\cdots\!20}a^{28}-\frac{67\!\cdots\!59}{25\!\cdots\!20}a^{27}+\frac{10\!\cdots\!23}{42\!\cdots\!20}a^{26}-\frac{41\!\cdots\!23}{25\!\cdots\!20}a^{25}+\frac{11\!\cdots\!59}{25\!\cdots\!20}a^{24}-\frac{19\!\cdots\!63}{42\!\cdots\!20}a^{23}+\frac{71\!\cdots\!43}{25\!\cdots\!20}a^{22}-\frac{10\!\cdots\!19}{25\!\cdots\!20}a^{21}-\frac{16\!\cdots\!57}{42\!\cdots\!20}a^{20}-\frac{21\!\cdots\!43}{25\!\cdots\!20}a^{19}-\frac{11\!\cdots\!81}{25\!\cdots\!20}a^{18}-\frac{10\!\cdots\!63}{42\!\cdots\!20}a^{17}-\frac{73\!\cdots\!97}{11\!\cdots\!20}a^{16}+\frac{52\!\cdots\!01}{11\!\cdots\!20}a^{15}+\frac{23\!\cdots\!83}{19\!\cdots\!20}a^{14}+\frac{18\!\cdots\!77}{55\!\cdots\!20}a^{13}-\frac{77\!\cdots\!01}{55\!\cdots\!20}a^{12}+\frac{22\!\cdots\!87}{91\!\cdots\!20}a^{11}+\frac{14\!\cdots\!83}{15\!\cdots\!20}a^{10}+\frac{11\!\cdots\!11}{25\!\cdots\!20}a^{9}-\frac{58\!\cdots\!97}{11\!\cdots\!20}a^{8}+\frac{34\!\cdots\!07}{70\!\cdots\!20}a^{7}+\frac{56\!\cdots\!19}{11\!\cdots\!20}a^{6}+\frac{226852560493853}{546089094042120}a^{5}-\frac{13\!\cdots\!63}{32\!\cdots\!20}a^{4}+\frac{258148281924317}{546089094042120}a^{3}+\frac{1270008728177}{5056380500390}a^{2}-\frac{2861580985633}{7584570750585}a-\frac{1042485312378}{2528190250195}$, $\frac{1}{15\!\cdots\!20}a^{39}-\frac{1}{15\!\cdots\!20}a^{38}+\frac{1}{25\!\cdots\!20}a^{37}-\frac{17}{15\!\cdots\!20}a^{36}+\frac{17}{15\!\cdots\!20}a^{35}-\frac{17}{25\!\cdots\!20}a^{34}+\frac{73}{15\!\cdots\!20}a^{33}-\frac{73}{15\!\cdots\!20}a^{32}+\frac{73}{25\!\cdots\!20}a^{31}+\frac{413201720782207}{15\!\cdots\!20}a^{30}+\frac{10\!\cdots\!93}{15\!\cdots\!20}a^{29}-\frac{78\!\cdots\!23}{15\!\cdots\!20}a^{28}+\frac{21\!\cdots\!63}{25\!\cdots\!20}a^{27}-\frac{60\!\cdots\!03}{15\!\cdots\!20}a^{26}+\frac{13\!\cdots\!83}{15\!\cdots\!20}a^{25}-\frac{38\!\cdots\!23}{25\!\cdots\!20}a^{24}+\frac{79\!\cdots\!63}{15\!\cdots\!20}a^{23}-\frac{57\!\cdots\!43}{15\!\cdots\!20}a^{22}+\frac{12\!\cdots\!23}{25\!\cdots\!20}a^{21}-\frac{67\!\cdots\!03}{15\!\cdots\!20}a^{20}-\frac{37\!\cdots\!97}{15\!\cdots\!20}a^{19}-\frac{37\!\cdots\!83}{25\!\cdots\!20}a^{18}-\frac{70\!\cdots\!17}{71\!\cdots\!20}a^{17}+\frac{33\!\cdots\!97}{71\!\cdots\!20}a^{16}-\frac{50\!\cdots\!97}{11\!\cdots\!20}a^{15}-\frac{11\!\cdots\!83}{33\!\cdots\!20}a^{14}-\frac{12\!\cdots\!77}{33\!\cdots\!20}a^{13}+\frac{27\!\cdots\!97}{55\!\cdots\!20}a^{12}+\frac{29\!\cdots\!93}{15\!\cdots\!20}a^{11}-\frac{60\!\cdots\!23}{15\!\cdots\!20}a^{10}-\frac{42\!\cdots\!31}{21\!\cdots\!60}a^{9}+\frac{83\!\cdots\!77}{42\!\cdots\!20}a^{8}-\frac{13\!\cdots\!47}{70\!\cdots\!20}a^{7}+\frac{18\!\cdots\!99}{98\!\cdots\!60}a^{6}-\frac{33\!\cdots\!13}{19\!\cdots\!20}a^{5}+\frac{390290038496083}{32\!\cdots\!20}a^{4}-\frac{11335205325797}{45507424503510}a^{3}+\frac{23884678628309}{91014849007020}a^{2}-\frac{900958631459}{15169141501170}a-\frac{213464336940}{505638050039}$ 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, 1/6*a^21 - 1/6*a^20 + 1/6*a^18 - 1/6*a^17 + 1/6*a^15 - 1/6*a^14 + 1/6*a^12 - 1/6*a^11 + 1/6*a^10 - 1/6*a^8 + 1/6*a^7 - 1/6*a^5 + 1/6*a^4 - 1/6*a^2 + 1/6*a, 1/182124*a^22 - 1/36*a^21 - 1/3*a^20 + 1/36*a^19 + 17/36*a^18 - 1/3*a^17 - 17/36*a^16 - 1/36*a^15 - 1/3*a^14 + 1/36*a^13 + 17/36*a^12 + 78907/182124*a^11 + 1/3*a^10 - 13/36*a^9 - 5/36*a^8 + 1/3*a^7 + 5/36*a^6 + 13/36*a^5 + 1/3*a^4 - 13/36*a^3 - 5/36*a^2 + 1/3*a + 168/5059, 1/1092744*a^23 - 1/1092744*a^22 + 1/36*a^21 + 1/216*a^20 + 107/216*a^19 - 17/36*a^18 - 17/216*a^17 - 91/216*a^16 + 1/36*a^15 + 73/216*a^14 + 35/216*a^13 + 352093/1092744*a^12 + 83813/182124*a^11 + 95/216*a^10 + 13/216*a^9 + 5/36*a^8 - 103/216*a^7 - 5/216*a^6 - 13/36*a^5 + 23/216*a^4 + 85/216*a^3 + 5/36*a^2 - 5003/10118*a - 28/5059, 1/6556464*a^24 - 1/6556464*a^23 + 1/1092744*a^22 - 107/1296*a^21 + 107/1296*a^20 - 107/216*a^19 + 307/1296*a^18 - 307/1296*a^17 + 91/216*a^16 - 251/1296*a^15 + 251/1296*a^14 + 3083953/6556464*a^13 + 539123/1092744*a^12 - 2172415/6556464*a^11 - 635/1296*a^10 - 13/216*a^9 - 211/1296*a^8 + 211/1296*a^7 + 5/216*a^6 - 85/1296*a^5 + 85/1296*a^4 - 85/216*a^3 - 1677/10118*a^2 + 1677/10118*a + 28/5059, 1/39338784*a^25 - 1/39338784*a^24 + 1/6556464*a^23 - 17/39338784*a^22 + 539/7776*a^21 - 107/1296*a^20 + 2467/7776*a^19 + 2717/7776*a^18 + 307/1296*a^17 - 3275/7776*a^16 + 683/7776*a^15 - 3472511/39338784*a^14 - 3103357/6556464*a^13 - 13099855/39338784*a^12 + 13137439/39338784*a^11 + 635/1296*a^10 + 1517/7776*a^9 + 1075/7776*a^8 - 211/1296*a^7 + 347/7776*a^6 + 2245/7776*a^5 - 85/1296*a^4 + 65795/182124*a^3 - 5087/182124*a^2 - 1677/10118*a + 1111/5059, 1/236032704*a^26 - 1/236032704*a^25 + 1/39338784*a^24 - 17/236032704*a^23 + 17/236032704*a^22 - 539/7776*a^21 + 10243/46656*a^20 + 20861/46656*a^19 - 2717/7776*a^18 - 11051/46656*a^17 - 4501/46656*a^16 + 101430913/236032704*a^15 + 3453107/39338784*a^14 + 39351857/236032704*a^13 + 39363295/236032704*a^12 - 12989695/39338784*a^11 - 21811/46656*a^10 - 9293/46656*a^9 - 1075/7776*a^8 + 347/46656*a^7 + 15205/46656*a^6 - 2245/7776*a^5 - 116329/1092744*a^4 + 480577/1092744*a^3 + 5087/182124*a^2 + 3085/15177*a + 658/5059, 1/1416196224*a^27 - 1/1416196224*a^26 + 1/236032704*a^25 - 17/1416196224*a^24 + 17/1416196224*a^23 - 17/236032704*a^22 - 20861/279936*a^21 + 20861/279936*a^20 - 20861/46656*a^19 - 88811/279936*a^18 + 88811/279936*a^17 - 449312063/1416196224*a^16 - 22772749/236032704*a^15 + 39351857/1416196224*a^14 - 39314273/1416196224*a^13 + 39462017/236032704*a^12 + 870503/1416196224*a^11 - 55949/279936*a^10 + 9293/46656*a^9 - 61861/279936*a^8 + 61861/279936*a^7 - 15205/46656*a^6 + 2797655/6556464*a^5 - 2797655/6556464*a^4 - 480577/1092744*a^3 + 14519/30354*a^2 - 14519/30354*a - 658/5059, 1/8497177344*a^28 - 1/8497177344*a^27 + 1/1416196224*a^26 - 17/8497177344*a^25 + 17/8497177344*a^24 - 17/1416196224*a^23 + 73/8497177344*a^22 + 20861/1679616*a^21 + 25795/279936*a^20 + 750997/1679616*a^19 + 648683/1679616*a^18 - 3281704511/8497177344*a^17 + 449292659/1416196224*a^16 + 2871744305/8497177344*a^15 - 2871706721/8497177344*a^14 + 39462017/1416196224*a^13 - 1415325721/8497177344*a^12 + 1418405017/8497177344*a^11 + 9293/279936*a^10 - 341797/1679616*a^9 + 621733/1679616*a^8 - 108517/279936*a^7 + 9354119/39338784*a^6 - 2797655/39338784*a^5 + 1704911/6556464*a^4 - 15835/182124*a^3 + 46189/182124*a^2 + 4730/15177*a + 591/5059, 1/50983064064*a^29 - 1/50983064064*a^28 + 1/8497177344*a^27 - 17/50983064064*a^26 + 17/50983064064*a^25 - 17/8497177344*a^24 + 73/50983064064*a^23 - 73/50983064064*a^22 - 20861/1679616*a^21 - 928619/10077696*a^20 - 750997/10077696*a^19 + 12296453953/50983064064*a^18 + 3281685107/8497177344*a^17 - 21203591503/50983064064*a^16 - 21282257633/50983064064*a^15 + 2871854465/8497177344*a^14 + 8498047847/50983064064*a^13 + 2208793/50983064064*a^12 - 1416725401/8497177344*a^11 - 341797/10077696*a^10 + 2021413/10077696*a^9 - 621733/1679616*a^8 + 88031687/236032704*a^7 - 48692903/236032704*a^6 + 2797655/39338784*a^5 - 15835/1092744*a^4 + 197959/1092744*a^3 - 46189/182124*a^2 - 9527/30354*a + 2431/5059, 1/1529491921920*a^30 + 1/305898384384*a^29 + 19/1529491921920*a^27 - 17/305898384384*a^26 - 539/1529491921920*a^24 + 73/305898384384*a^23 + 1/302330880*a^21 + 10828693/60466176*a^20 + 6397315547/305898384384*a^19 - 539/7080981120*a^18 + 1816723/60466176*a^17 + 121548995393/305898384384*a^16 + 19/32782320*a^15 - 11701115/60466176*a^14 - 83270851609/305898384384*a^13 + 1/151770*a^12 - 12094685/60466176*a^11 - 12099109/60466176*a^10 + 1/5*a^9 - 1703594945/8497177344*a^8 + 284725607/1416196224*a^7 - 1/5*a^6 + 8189737/39338784*a^5 - 1290703/6556464*a^4 + 1/5*a^3 - 42509/182124*a^2 - 2431/30354*a - 1/5, 1/917219979778649425920*a^31 - 125315581/917219979778649425920*a^30 + 88014187/10191333108651660288*a^29 - 8547854741/917219979778649425920*a^28 + 48906132341/917219979778649425920*a^27 - 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1/256762892259316005694341120*a^37 + 1/42793815376552667615723520*a^36 - 17/256762892259316005694341120*a^35 + 17/256762892259316005694341120*a^34 - 17/42793815376552667615723520*a^33 + 73/256762892259316005694341120*a^32 - 73/256762892259316005694341120*a^31 - 2061090105911/42793815376552667615723520*a^30 + 417585930048127/256762892259316005694341120*a^29 - 479418633227647/256762892259316005694341120*a^28 - 67401082810541759/256762892259316005694341120*a^27 + 10490611587831923/42793815376552667615723520*a^26 - 412105665262067023/256762892259316005694341120*a^25 + 1148489580556470559/256762892259316005694341120*a^24 - 193373490474875263/42793815376552667615723520*a^23 + 7109350734232311143/256762892259316005694341120*a^22 - 10125365117049635305695719/256762892259316005694341120*a^21 - 16406127644925176213442457/42793815376552667615723520*a^20 - 21661365317900369731037743/256762892259316005694341120*a^19 - 114506160122494189148662481/256762892259316005694341120*a^18 - 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17/1540577353555896034166046720*a^35 - 17/256762892259316005694341120*a^34 + 73/1540577353555896034166046720*a^33 - 73/1540577353555896034166046720*a^32 + 73/256762892259316005694341120*a^31 + 413201720782207/1540577353555896034166046720*a^30 + 10740723179800193/1540577353555896034166046720*a^29 - 78717344913714623/1540577353555896034166046720*a^28 + 21656958414667763/256762892259316005694341120*a^27 - 602848076083540303/1540577353555896034166046720*a^26 + 1338194863533053983/1540577353555896034166046720*a^25 - 383043554997511423/256762892259316005694341120*a^24 + 7928421086583343463/1540577353555896034166046720*a^23 - 5746366178699559143/1540577353555896034166046720*a^22 + 12586915706205547338498023/256762892259316005694341120*a^21 - 670253168271576957658547503/1540577353555896034166046720*a^20 - 370348595082102032380006097/1540577353555896034166046720*a^19 - 37773132002641438871764783/256762892259316005694341120*a^18 - 707932044023826356760217/7132302562758777935953920*a^17 + 3341145059282173380732697/7132302562758777935953920*a^16 - 508540082795510229915097/1188717093793129655992320*a^15 - 11171449096577204296783/33019919272031379333120*a^14 - 1260109662412704974177/33019919272031379333120*a^13 + 270123852930815725097/5503319878671896555520*a^12 + 29207677411072165793/152869996629774904320*a^11 - 60211780928021333423/152869996629774904320*a^10 - 421415135714216231/2123194397635762560*a^9 + 838424953856183177/4246388795271525120*a^8 - 134225171595894347/707731465878587520*a^7 + 1804670138485999/9829603692758160*a^6 - 3365632295249713/19659207385516320*a^5 + 390290038496083/3276534564252720*a^4 - 11335205325797/45507424503510*a^3 + 23884678628309/91014849007020*a^2 - 900958631459/15169141501170*a - 213464336940/505638050039

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:	$19$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( -\frac{218264857259}{308115470711179206833209344} a^{39} + \frac{2400913429849}{1540577353555896034166046720} a^{38} + \frac{3710502573403}{308115470711179206833209344} a^{36} + \frac{6329680860511}{1540577353555896034166046720} a^{35} - \frac{15933334579907}{308115470711179206833209344} a^{33} - \frac{626201875476071}{1540577353555896034166046720} a^{32} + \frac{43229811565271}{1540577353555896034166046720} a^{30} + \frac{9278220817222831}{1540577353555896034166046720} a^{29} + \frac{6410220592839571}{308115470711179206833209344} a^{28} - \frac{6329680860511}{1188717093793129655992320} a^{27} - \frac{22470148789956791}{1540577353555896034166046720} a^{26} - \frac{108973750078272707}{308115470711179206833209344} a^{25} - \frac{2400913429849}{5503319878671896555520} a^{24} - \frac{1622103167090866049}{1540577353555896034166046720} a^{23} + \frac{467946103277288683}{308115470711179206833209344} a^{22} + \frac{218264857259}{25478332771629150720} a^{21} + \frac{6410220592839571}{304522109815357982638080} a^{20} - \frac{3961389797343948563}{308115470711179206833209344} a^{19} - \frac{218264857259}{4246388795271525120} a^{18} - \frac{3455108899540528769}{7132302562758777935953920} a^{17} - \frac{97241140941172421}{237743418758625931198464} a^{16} - \frac{4147032287921}{4246388795271525120} a^{15} + \frac{121794191263951849}{33019919272031379333120} a^{14} + \frac{12461832025202605}{1100663975734379311104} a^{13} + \frac{117644758062601}{4246388795271525120} a^{12} + \frac{6410220592839571}{152869996629774904320} a^{11} - \frac{530601867996629}{5095666554325830144} a^{10} - \frac{218264857259}{839373155815680} a^{9} - \frac{1605271176}{2528190250195} a^{8} - \frac{15933334579907}{23591048862619584} a^{7} + \frac{117644758062601}{19659207385516320} a^{6} + \frac{3710502573403}{109217818808424} a^{4} - \frac{4147032287921}{91014849007020} a^{3} - \frac{218264857259}{505638050039} a - \frac{1309589143554}{2528190250195} \) -(218264857259)/(308115470711179206833209344)a^(39) + (2400913429849)/(1540577353555896034166046720)a^(38) + (3710502573403)/(308115470711179206833209344)a^(36) + (6329680860511)/(1540577353555896034166046720)a^(35) - (15933334579907)/(308115470711179206833209344)a^(33) - (626201875476071)/(1540577353555896034166046720)a^(32) + (43229811565271)/(1540577353555896034166046720)a^(30) + (9278220817222831)/(1540577353555896034166046720)a^(29) + (6410220592839571)/(308115470711179206833209344)a^(28) - (6329680860511)/(1188717093793129655992320)a^(27) - (22470148789956791)/(1540577353555896034166046720)a^(26) - (108973750078272707)/(308115470711179206833209344)a^(25) - (2400913429849)/(5503319878671896555520)a^(24) - (1622103167090866049)/(1540577353555896034166046720)a^(23) + (467946103277288683)/(308115470711179206833209344)a^(22) + (218264857259)/(25478332771629150720)a^(21) + (6410220592839571)/(304522109815357982638080)a^(20) - (3961389797343948563)/(308115470711179206833209344)a^(19) - (218264857259)/(4246388795271525120)a^(18) - (3455108899540528769)/(7132302562758777935953920)a^(17) - (97241140941172421)/(237743418758625931198464)a^(16) - (4147032287921)/(4246388795271525120)a^(15) + (121794191263951849)/(33019919272031379333120)a^(14) + (12461832025202605)/(1100663975734379311104)a^(13) + (117644758062601)/(4246388795271525120)a^(12) + (6410220592839571)/(152869996629774904320)a^(11) - (530601867996629)/(5095666554325830144)a^(10) - (218264857259)/(839373155815680)a^(9) - (1605271176)/(2528190250195)a^(8) - (15933334579907)/(23591048862619584)a^(7) + (117644758062601)/(19659207385516320)a^(6) + (3710502573403)/(109217818808424)a^(4) - (4147032287921)/(91014849007020)a^(3) - (218264857259)/(505638050039)a - (1309589143554)/(2528190250195) (order $66$)	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^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976)

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^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976, 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^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976);

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^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976);

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

$C_2^2\times C_{10}$ (as 40T7):

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)

An abelian group of order 40

The 40 conjugacy class representatives for $C_2^2\times C_{10}$

Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

$\Q(\sqrt{-23}) $, $\Q(\sqrt{33}) $, $\Q(\sqrt{-759}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{69}) $, $\Q(\sqrt{-11}) $, $\Q(\sqrt{253}) $, $\Q(\sqrt{-23}, \sqrt{33})$, $\Q(\sqrt{-3}, \sqrt{-23})$, $\Q(\sqrt{-11}, \sqrt{-23})$, $\Q(\sqrt{-3}, \sqrt{-11})$, $\Q(\sqrt{33}, \sqrt{69})$, $\Q(\sqrt{-3}, \sqrt{253})$, $\Q(\sqrt{-11}, \sqrt{69})$, $\Q(\zeta_{11})^+$, 8.0.331869318561.5, 10.0.1379687283212183.1, $\Q(\zeta_{33})^+$, 10.0.3687904108026165159.1, 10.0.52089208083.1, 10.10.335264009820560469.1, $\Q(\zeta_{11})$, 10.10.15176560115334013.1, 20.0.13600636709996264858725830959545495281.1, 20.0.112401956280960866601039925285499961.1, 20.0.230327976934347149972494554684169.1, $\Q(\zeta_{33})$, 20.20.13600636709996264858725830959545495281.1, 20.0.13600636709996264858725830959545495281.2, 20.0.13600636709996264858725830959545495281.3

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	${\href{/padicField/2.10.0.1}{10} }^{4}$	R	${\href{/padicField/5.10.0.1}{10} }^{4}$	${\href{/padicField/7.10.0.1}{10} }^{4}$	R	${\href{/padicField/13.10.0.1}{10} }^{4}$	${\href{/padicField/17.10.0.1}{10} }^{4}$	${\href{/padicField/19.10.0.1}{10} }^{4}$	R	${\href{/padicField/29.10.0.1}{10} }^{4}$	${\href{/padicField/31.5.0.1}{5} }^{8}$	${\href{/padicField/37.10.0.1}{10} }^{4}$	${\href{/padicField/41.10.0.1}{10} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{20}$	${\href{/padicField/47.10.0.1}{10} }^{4}$	${\href{/padicField/53.10.0.1}{10} }^{4}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3	3.10.5.2	$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
	3.10.5.2	$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
	3.10.5.2	$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
	3.10.5.2	$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$
$23$ 23	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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