LMFDB - Number field 45.45.145251134032895891800542990495810490845564676151386135305167614095927459824506204505456009867803146562437177635729312896728515625.2

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651)

gp: K = bnfinit(y^45 - 405*y^43 - 45*y^42 + 74565*y^41 + 13986*y^40 - 8290800*y^39 - 1937610*y^38 + 623822085*y^37 + 158343200*y^36 - 33730092612*y^35 - 8501973165*y^34 + 1358280567105*y^33 + 315527932755*y^32 - 41656181730075*y^31 - 8257524539733*y^30 + 986564158362990*y^29 + 151291718024085*y^28 - 18188682238507875*y^27 - 1837307213515305*y^26 + 261951082234160256*y^25 + 11633106453685350*y^24 - 2946259043332552305*y^23 + 39719938240377450*y^22 + 25786532715466797195*y^21 - 1805631856520318496*y^20 - 174433501954113158925*y^19 + 21474396909776230700*y^18 + 902723065221863783160*y^17 - 154718694923268688020*y^16 - 3523794539226841295061*y^15 + 755660668535950676940*y^14 + 10175405955078415341510*y^13 - 2572756413109899568440*y^12 - 21151381993350223213305*y^11 + 6092956812785525217882*y^10 + 30403883242511231332790*y^9 - 9799931675663506270395*y^8 - 28334864017939625412120*y^7 + 10152119383093656646320*y^6 + 15176238217723477388907*y^5 - 6060909987353410350210*y^4 - 3421782109383146916285*y^3 + 1568220560944780065255*y^2 - 73641665210792649300*y - 1914182976156359651, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651)

$ x^{45} - 405 x^{43} - 45 x^{42} + 74565 x^{41} + 13986 x^{40} - 8290800 x^{39} - 1937610 x^{38} + \cdots - 19\!\cdots\!51 $ x^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$45$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[45, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$145\!\cdots\!625$ 145251134032895891800542990495810490845564676151386135305167614095927459824506204505456009867803146562437177635729312896728515625 $\medspace = 3^{110}\cdot 5^{72}\cdot 7^{30}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$704.77$	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:	$3^{22/9}5^{8/5}7^{2/3}\approx 704.769499459809$
Ramified primes:	$3$, $5$, $7$ 3, 5, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$45$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$4725=3^{3}\cdot 5^{2}\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{4725}(4096,·)$, $\chi_{4725}(1,·)$, $\chi_{4725}(3586,·)$, $\chi_{4725}(1411,·)$, $\chi_{4725}(4246,·)$, $\chi_{4725}(3466,·)$, $\chi_{4725}(2956,·)$, $\chi_{4725}(781,·)$, $\chi_{4725}(2836,·)$, $\chi_{4725}(2326,·)$, $\chi_{4725}(151,·)$, $\chi_{4725}(2671,·)$, $\chi_{4725}(2206,·)$, $\chi_{4725}(1696,·)$, $\chi_{4725}(3931,·)$, $\chi_{4725}(1576,·)$, $\chi_{4725}(1066,·)$, $\chi_{4725}(946,·)$, $\chi_{4725}(4531,·)$, $\chi_{4725}(2356,·)$, $\chi_{4725}(436,·)$, $\chi_{4725}(4411,·)$, $\chi_{4725}(316,·)$, $\chi_{4725}(3901,·)$, $\chi_{4725}(1726,·)$, $\chi_{4725}(3616,·)$, $\chi_{4725}(3781,·)$, $\chi_{4725}(3271,·)$, $\chi_{4725}(1096,·)$, $\chi_{4725}(3151,·)$, $\chi_{4725}(2641,·)$, $\chi_{4725}(466,·)$, $\chi_{4725}(121,·)$, $\chi_{4725}(2521,·)$, $\chi_{4725}(2011,·)$, $\chi_{4725}(3301,·)$, $\chi_{4725}(1891,·)$, $\chi_{4725}(1381,·)$, $\chi_{4725}(4561,·)$, $\chi_{4725}(1261,·)$, $\chi_{4725}(751,·)$, $\chi_{4725}(631,·)$, $\chi_{4725}(4216,·)$, $\chi_{4725}(2041,·)$, $\chi_{4725}(2986,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{2}{7}a^{11}+\frac{3}{7}a^{10}-\frac{1}{7}a^{9}-\frac{3}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}-\frac{2}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{13}-\frac{1}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}-\frac{3}{7}a^{7}-\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{11}-\frac{2}{7}a^{10}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{15}-\frac{3}{7}a^{11}+\frac{2}{7}a^{10}-\frac{3}{7}a^{9}+\frac{3}{7}a^{8}+\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{16}+\frac{3}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{7}+\frac{2}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{17}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{3}{7}a^{9}-\frac{1}{7}a^{8}-\frac{3}{7}a^{7}-\frac{3}{7}a^{6}-\frac{1}{7}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{18}+\frac{1}{7}a^{11}+\frac{2}{7}a^{10}-\frac{3}{7}a^{9}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{19}-\frac{3}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{20}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{21}-\frac{2}{7}a^{11}-\frac{1}{7}a^{10}+\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{22}+\frac{2}{7}a^{11}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{2}{7}a^{2}-\frac{1}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{23}-\frac{3}{7}a^{11}+\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{3}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a-\frac{2}{7}$, $\frac{1}{49}a^{24}+\frac{3}{49}a^{23}+\frac{3}{49}a^{22}+\frac{1}{49}a^{18}+\frac{1}{49}a^{16}-\frac{2}{49}a^{15}+\frac{1}{49}a^{12}-\frac{1}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{49}a^{9}-\frac{17}{49}a^{8}-\frac{1}{7}a^{7}+\frac{1}{49}a^{6}+\frac{1}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{2}{49}a^{2}+\frac{1}{49}a+\frac{1}{49}$, $\frac{1}{931}a^{25}-\frac{2}{931}a^{24}+\frac{16}{931}a^{23}-\frac{43}{931}a^{22}+\frac{3}{133}a^{21}-\frac{4}{133}a^{20}-\frac{55}{931}a^{19}-\frac{26}{931}a^{18}+\frac{15}{931}a^{17}-\frac{2}{133}a^{16}-\frac{32}{931}a^{15}+\frac{8}{133}a^{14}+\frac{22}{931}a^{13}-\frac{40}{931}a^{12}-\frac{46}{133}a^{11}-\frac{327}{931}a^{10}-\frac{48}{931}a^{9}-\frac{384}{931}a^{8}+\frac{183}{931}a^{7}+\frac{345}{931}a^{6}+\frac{11}{133}a^{5}+\frac{53}{133}a^{4}+\frac{201}{931}a^{3}-\frac{2}{49}a^{2}+\frac{185}{931}a+\frac{149}{931}$, $\frac{1}{931}a^{26}-\frac{1}{133}a^{24}+\frac{65}{931}a^{23}+\frac{11}{931}a^{22}+\frac{2}{133}a^{21}+\frac{22}{931}a^{20}-\frac{3}{931}a^{19}-\frac{8}{133}a^{18}+\frac{16}{931}a^{17}+\frac{54}{931}a^{16}+\frac{30}{931}a^{15}+\frac{1}{931}a^{14}+\frac{4}{931}a^{13}-\frac{22}{931}a^{12}-\frac{40}{931}a^{11}+\frac{229}{931}a^{10}+\frac{40}{133}a^{9}-\frac{129}{931}a^{8}+\frac{46}{931}a^{7}-\frac{316}{931}a^{6}+\frac{8}{19}a^{5}+\frac{145}{931}a^{4}-\frac{62}{133}a^{3}-\frac{17}{133}a^{2}-\frac{32}{931}a+\frac{279}{931}$, $\frac{1}{931}a^{27}-\frac{6}{931}a^{24}-\frac{48}{931}a^{23}-\frac{59}{931}a^{22}+\frac{36}{931}a^{21}-\frac{66}{931}a^{20}-\frac{6}{133}a^{19}+\frac{43}{931}a^{18}+\frac{26}{931}a^{17}+\frac{8}{931}a^{16}+\frac{24}{931}a^{15}-\frac{3}{931}a^{14}-\frac{1}{931}a^{13}+\frac{22}{931}a^{12}-\frac{30}{931}a^{11}+\frac{55}{133}a^{10}+\frac{352}{931}a^{9}-\frac{30}{133}a^{8}+\frac{34}{931}a^{7}-\frac{176}{931}a^{6}+\frac{1}{49}a^{5}-\frac{2}{19}a^{4}+\frac{32}{133}a^{3}+\frac{215}{931}a^{2}-\frac{79}{931}a-\frac{78}{931}$, $\frac{1}{931}a^{28}-\frac{3}{931}a^{24}-\frac{58}{931}a^{23}-\frac{51}{931}a^{22}+\frac{60}{931}a^{21}+\frac{8}{133}a^{20}-\frac{3}{133}a^{19}+\frac{60}{931}a^{18}-\frac{5}{133}a^{17}-\frac{3}{931}a^{16}-\frac{43}{931}a^{15}-\frac{64}{931}a^{14}+\frac{3}{133}a^{13}+\frac{53}{931}a^{12}+\frac{45}{133}a^{11}+\frac{36}{133}a^{10}+\frac{148}{931}a^{9}+\frac{86}{931}a^{8}+\frac{124}{931}a^{7}+\frac{417}{931}a^{6}-\frac{5}{133}a^{5}-\frac{11}{133}a^{4}-\frac{9}{19}a^{3}+\frac{244}{931}a^{2}-\frac{374}{931}a-\frac{379}{931}$, $\frac{1}{931}a^{29}-\frac{1}{133}a^{24}+\frac{5}{133}a^{23}-\frac{31}{931}a^{22}-\frac{2}{133}a^{21}+\frac{4}{133}a^{20}+\frac{4}{133}a^{19}-\frac{8}{133}a^{18}+\frac{6}{133}a^{17}-\frac{4}{133}a^{16}-\frac{8}{931}a^{15}+\frac{8}{133}a^{14}-\frac{2}{133}a^{13}-\frac{2}{133}a^{12}-\frac{64}{133}a^{11}+\frac{33}{133}a^{10}-\frac{30}{133}a^{9}-\frac{135}{931}a^{8}-\frac{52}{133}a^{7}-\frac{1}{133}a^{6}-\frac{54}{133}a^{5}+\frac{58}{133}a^{4}+\frac{45}{133}a^{3}-\frac{10}{133}a^{2}-\frac{166}{931}a-\frac{61}{133}$, $\frac{1}{931}a^{30}+\frac{2}{931}a^{24}+\frac{24}{931}a^{23}+\frac{27}{931}a^{22}+\frac{6}{133}a^{21}-\frac{5}{133}a^{20}-\frac{6}{133}a^{19}-\frac{26}{931}a^{18}-\frac{8}{133}a^{17}+\frac{8}{931}a^{16}+\frac{3}{931}a^{15}-\frac{3}{133}a^{14}+\frac{1}{133}a^{13}+\frac{51}{931}a^{12}-\frac{6}{19}a^{11}-\frac{53}{133}a^{10}+\frac{422}{931}a^{9}-\frac{335}{931}a^{8}+\frac{11}{133}a^{7}-\frac{376}{931}a^{6}-\frac{55}{133}a^{5}+\frac{55}{133}a^{4}+\frac{58}{133}a^{3}+\frac{5}{931}a^{2}+\frac{184}{931}a+\frac{93}{931}$, $\frac{1}{931}a^{31}+\frac{9}{931}a^{24}-\frac{62}{931}a^{23}-\frac{62}{931}a^{22}+\frac{8}{133}a^{21}+\frac{2}{133}a^{20}-\frac{1}{19}a^{19}-\frac{23}{931}a^{18}-\frac{22}{931}a^{17}+\frac{12}{931}a^{16}-\frac{52}{931}a^{15}+\frac{4}{133}a^{14}+\frac{1}{133}a^{13}+\frac{33}{931}a^{12}-\frac{8}{19}a^{11}+\frac{278}{931}a^{10}-\frac{277}{931}a^{9}+\frac{237}{931}a^{8}+\frac{46}{133}a^{7}-\frac{163}{931}a^{6}+\frac{52}{133}a^{5}+\frac{66}{133}a^{4}+\frac{2}{931}a^{3}+\frac{431}{931}a^{2}-\frac{30}{931}a+\frac{82}{931}$, $\frac{1}{931}a^{32}-\frac{6}{931}a^{24}+\frac{41}{931}a^{23}+\frac{25}{931}a^{22}-\frac{6}{133}a^{21}-\frac{9}{133}a^{20}-\frac{60}{931}a^{19}-\frac{16}{931}a^{18}+\frac{10}{931}a^{17}-\frac{3}{133}a^{16}-\frac{26}{931}a^{15}+\frac{5}{133}a^{14}-\frac{32}{931}a^{13}+\frac{6}{931}a^{12}-\frac{149}{931}a^{11}+\frac{6}{931}a^{10}+\frac{80}{931}a^{9}-\frac{193}{931}a^{8}+\frac{185}{931}a^{7}-\frac{176}{931}a^{6}+\frac{43}{133}a^{5}+\frac{387}{931}a^{4}-\frac{314}{931}a^{3}+\frac{103}{931}a^{2}+\frac{184}{931}a-\frac{106}{931}$, $\frac{1}{931}a^{33}-\frac{9}{931}a^{24}+\frac{1}{133}a^{23}-\frac{15}{931}a^{22}+\frac{9}{133}a^{21}+\frac{2}{49}a^{20}+\frac{53}{931}a^{19}-\frac{51}{931}a^{18}-\frac{64}{931}a^{17}-\frac{15}{931}a^{16}+\frac{52}{931}a^{15}+\frac{2}{49}a^{14}+\frac{5}{931}a^{13}-\frac{4}{133}a^{12}+\frac{202}{931}a^{11}-\frac{153}{931}a^{10}+\frac{241}{931}a^{9}-\frac{409}{931}a^{8}-\frac{408}{931}a^{7}+\frac{205}{931}a^{6}-\frac{215}{931}a^{5}-\frac{216}{931}a^{4}-\frac{60}{133}a^{3}+\frac{298}{931}a^{2}+\frac{43}{133}a-\frac{208}{931}$, $\frac{1}{931}a^{34}+\frac{8}{931}a^{24}+\frac{53}{931}a^{23}-\frac{1}{931}a^{22}-\frac{39}{931}a^{21}-\frac{66}{931}a^{20}-\frac{2}{133}a^{19}-\frac{13}{931}a^{18}-\frac{13}{931}a^{17}-\frac{55}{931}a^{16}-\frac{22}{931}a^{15}-\frac{23}{931}a^{14}+\frac{37}{931}a^{13}-\frac{6}{931}a^{12}-\frac{391}{931}a^{11}+\frac{13}{133}a^{10}-\frac{138}{931}a^{9}-\frac{463}{931}a^{8}-\frac{276}{931}a^{7}+\frac{249}{931}a^{6}-\frac{321}{931}a^{5}-\frac{58}{133}a^{4}-\frac{3}{133}a^{3}-\frac{212}{931}a^{2}+\frac{412}{931}a+\frac{163}{931}$, $\frac{1}{931}a^{35}-\frac{1}{133}a^{24}+\frac{6}{133}a^{23}-\frac{8}{133}a^{22}+\frac{32}{931}a^{21}-\frac{8}{133}a^{20}+\frac{4}{133}a^{19}-\frac{2}{133}a^{18}-\frac{6}{133}a^{17}+\frac{2}{133}a^{16}-\frac{2}{133}a^{15}-\frac{12}{931}a^{14}-\frac{1}{19}a^{13}-\frac{2}{133}a^{12}-\frac{37}{133}a^{11}-\frac{64}{133}a^{10}-\frac{2}{19}a^{9}+\frac{2}{19}a^{8}-\frac{284}{931}a^{7}+\frac{24}{133}a^{6}+\frac{9}{19}a^{5}-\frac{47}{133}a^{4}+\frac{9}{19}a^{3}-\frac{47}{133}a^{2}-\frac{66}{133}a-\frac{71}{931}$, $\frac{1}{6517}a^{36}+\frac{1}{6517}a^{35}+\frac{2}{6517}a^{33}+\frac{3}{6517}a^{32}-\frac{1}{6517}a^{31}+\frac{3}{6517}a^{29}+\frac{1}{6517}a^{28}+\frac{1}{6517}a^{26}-\frac{2}{6517}a^{25}-\frac{32}{6517}a^{24}+\frac{62}{931}a^{23}+\frac{457}{6517}a^{22}+\frac{24}{343}a^{21}+\frac{1}{49}a^{20}+\frac{40}{6517}a^{19}-\frac{458}{6517}a^{18}+\frac{211}{6517}a^{17}-\frac{65}{931}a^{16}+\frac{349}{6517}a^{15}-\frac{454}{6517}a^{14}+\frac{10}{931}a^{13}-\frac{403}{6517}a^{12}+\frac{2416}{6517}a^{11}+\frac{1486}{6517}a^{10}-\frac{390}{931}a^{9}-\frac{2915}{6517}a^{8}-\frac{839}{6517}a^{7}+\frac{202}{931}a^{6}-\frac{72}{343}a^{5}-\frac{71}{6517}a^{4}-\frac{1927}{6517}a^{3}+\frac{81}{931}a^{2}+\frac{1090}{6517}a-\frac{41}{6517}$, $\frac{1}{45619}a^{37}-\frac{3}{45619}a^{36}+\frac{3}{45619}a^{35}-\frac{12}{45619}a^{34}-\frac{5}{45619}a^{33}-\frac{6}{45619}a^{32}-\frac{17}{45619}a^{31}-\frac{4}{45619}a^{30}-\frac{11}{45619}a^{29}-\frac{11}{45619}a^{28}+\frac{1}{45619}a^{27}-\frac{20}{45619}a^{26}-\frac{3}{45619}a^{25}-\frac{33}{45619}a^{24}+\frac{590}{45619}a^{23}-\frac{2}{931}a^{22}+\frac{2726}{45619}a^{21}-\frac{807}{45619}a^{20}+\frac{2504}{45619}a^{19}+\frac{1301}{45619}a^{18}-\frac{2013}{45619}a^{17}+\frac{3149}{45619}a^{16}+\frac{2714}{45619}a^{15}-\frac{2979}{45619}a^{14}+\frac{157}{45619}a^{13}+\frac{1837}{45619}a^{12}-\frac{6470}{45619}a^{11}-\frac{1186}{2401}a^{10}+\frac{20143}{45619}a^{9}-\frac{21848}{45619}a^{8}+\frac{7626}{45619}a^{7}-\frac{22095}{45619}a^{6}-\frac{17692}{45619}a^{5}+\frac{19763}{45619}a^{4}+\frac{21820}{45619}a^{3}-\frac{18496}{45619}a^{2}-\frac{11674}{45619}a-\frac{396}{45619}$, $\frac{1}{45619}a^{38}+\frac{1}{45619}a^{36}+\frac{4}{45619}a^{35}+\frac{8}{45619}a^{34}-\frac{1}{6517}a^{33}-\frac{2}{6517}a^{32}-\frac{13}{45619}a^{31}-\frac{23}{45619}a^{30}-\frac{23}{45619}a^{29}+\frac{24}{45619}a^{28}-\frac{17}{45619}a^{27}-\frac{1}{6517}a^{26}-\frac{1}{6517}a^{25}-\frac{419}{45619}a^{24}+\frac{137}{2401}a^{23}+\frac{2201}{45619}a^{22}+\frac{431}{6517}a^{21}+\frac{916}{45619}a^{20}+\frac{305}{6517}a^{19}+\frac{204}{6517}a^{18}+\frac{3193}{45619}a^{17}+\frac{1234}{45619}a^{16}-\frac{2880}{45619}a^{15}+\frac{978}{45619}a^{14}+\frac{740}{45619}a^{13}+\frac{384}{6517}a^{12}+\frac{45}{343}a^{11}+\frac{8121}{45619}a^{10}+\frac{9377}{45619}a^{9}-\frac{2327}{6517}a^{8}+\frac{7944}{45619}a^{7}-\frac{16357}{45619}a^{6}-\frac{2788}{6517}a^{5}+\frac{2878}{6517}a^{4}-\frac{482}{45619}a^{3}+\frac{5799}{45619}a^{2}-\frac{20830}{45619}a+\frac{5483}{45619}$, $\frac{1}{45619}a^{39}-\frac{2}{45619}a^{35}+\frac{5}{45619}a^{34}-\frac{23}{45619}a^{33}+\frac{3}{6517}a^{32}+\frac{1}{45619}a^{31}-\frac{1}{2401}a^{30}+\frac{2}{6517}a^{29}-\frac{13}{45619}a^{28}-\frac{8}{45619}a^{27}+\frac{6}{45619}a^{26}-\frac{10}{45619}a^{25}-\frac{80}{45619}a^{24}+\frac{1268}{45619}a^{23}-\frac{250}{6517}a^{22}+\frac{1172}{45619}a^{21}+\frac{982}{45619}a^{20}+\frac{212}{45619}a^{19}-\frac{1223}{45619}a^{18}+\frac{1623}{45619}a^{17}+\frac{1811}{45619}a^{16}-\frac{177}{6517}a^{15}-\frac{2021}{45619}a^{14}+\frac{2580}{45619}a^{13}+\frac{145}{2401}a^{12}-\frac{5506}{45619}a^{11}+\frac{575}{2401}a^{10}-\frac{3357}{45619}a^{9}+\frac{10}{6517}a^{8}-\frac{2479}{45619}a^{7}-\frac{12905}{45619}a^{6}+\frac{14143}{45619}a^{5}+\frac{753}{2401}a^{4}-\frac{17330}{45619}a^{3}-\frac{18945}{45619}a^{2}-\frac{3084}{6517}a+\frac{12933}{45619}$, $\frac{1}{319333}a^{40}-\frac{1}{319333}a^{39}+\frac{3}{319333}a^{38}+\frac{3}{319333}a^{37}+\frac{6}{319333}a^{36}-\frac{15}{45619}a^{35}+\frac{156}{319333}a^{34}-\frac{160}{319333}a^{33}+\frac{158}{319333}a^{32}+\frac{170}{319333}a^{31}+\frac{1}{319333}a^{30}+\frac{11}{319333}a^{29}+\frac{156}{319333}a^{28}-\frac{83}{319333}a^{27}+\frac{15}{319333}a^{26}+\frac{117}{319333}a^{25}-\frac{1044}{319333}a^{24}-\frac{3631}{319333}a^{23}-\frac{7450}{319333}a^{22}+\frac{962}{16807}a^{21}-\frac{884}{319333}a^{20}-\frac{4353}{319333}a^{19}+\frac{10795}{319333}a^{18}+\frac{13738}{319333}a^{17}-\frac{12588}{319333}a^{16}-\frac{11290}{319333}a^{15}+\frac{15811}{319333}a^{14}-\frac{6101}{319333}a^{13}-\frac{13999}{319333}a^{12}-\frac{8013}{16807}a^{11}-\frac{135158}{319333}a^{10}+\frac{7954}{45619}a^{9}-\frac{150322}{319333}a^{8}+\frac{101125}{319333}a^{7}-\frac{100754}{319333}a^{6}-\frac{144430}{319333}a^{5}-\frac{115617}{319333}a^{4}-\frac{145732}{319333}a^{3}+\frac{77160}{319333}a^{2}+\frac{1514}{319333}a-\frac{131722}{319333}$, $\frac{1}{35395827719}a^{41}-\frac{10016}{35395827719}a^{40}-\frac{266573}{35395827719}a^{39}-\frac{56495}{35395827719}a^{38}+\frac{218027}{35395827719}a^{37}+\frac{2634336}{35395827719}a^{36}+\frac{14698854}{35395827719}a^{35}+\frac{16095616}{35395827719}a^{34}+\frac{12828878}{35395827719}a^{33}-\frac{981285}{35395827719}a^{32}-\frac{11612911}{35395827719}a^{31}+\frac{4543076}{35395827719}a^{30}-\frac{9393430}{35395827719}a^{29}+\frac{18924008}{35395827719}a^{28}+\frac{8551301}{35395827719}a^{27}-\frac{2150758}{5056546817}a^{26}+\frac{7952345}{35395827719}a^{25}-\frac{28996876}{5056546817}a^{24}+\frac{414827345}{35395827719}a^{23}-\frac{1034265578}{35395827719}a^{22}+\frac{312654371}{5056546817}a^{21}+\frac{82412425}{35395827719}a^{20}+\frac{2214228201}{35395827719}a^{19}+\frac{1067322241}{35395827719}a^{18}-\frac{100037934}{1862938301}a^{17}-\frac{2016139506}{35395827719}a^{16}+\frac{1762389}{5056546817}a^{15}-\frac{57579278}{35395827719}a^{14}-\frac{45288890}{5056546817}a^{13}+\frac{2212086623}{35395827719}a^{12}+\frac{15361967849}{35395827719}a^{11}-\frac{15545469202}{35395827719}a^{10}-\frac{16227238444}{35395827719}a^{9}+\frac{7462660718}{35395827719}a^{8}+\frac{15988714913}{35395827719}a^{7}+\frac{8146565303}{35395827719}a^{6}-\frac{9186200075}{35395827719}a^{5}+\frac{6016756051}{35395827719}a^{4}-\frac{15644415058}{35395827719}a^{3}-\frac{489284102}{5056546817}a^{2}+\frac{10658644893}{35395827719}a+\frac{10596119759}{35395827719}$, $\frac{1}{35395827719}a^{42}-\frac{52228}{35395827719}a^{40}+\frac{156207}{35395827719}a^{39}-\frac{32250}{5056546817}a^{38}-\frac{317936}{35395827719}a^{37}-\frac{54872}{1862938301}a^{36}+\frac{7947124}{35395827719}a^{35}+\frac{17678807}{35395827719}a^{34}-\frac{1908673}{35395827719}a^{33}+\frac{9345509}{35395827719}a^{32}+\frac{5832154}{35395827719}a^{31}-\frac{8214987}{35395827719}a^{30}-\frac{6074560}{35395827719}a^{29}-\frac{11950642}{35395827719}a^{28}-\frac{18820211}{35395827719}a^{27}-\frac{10915057}{35395827719}a^{26}+\frac{1107223}{5056546817}a^{25}+\frac{113354790}{35395827719}a^{24}+\frac{279399327}{5056546817}a^{23}+\frac{56264451}{35395827719}a^{22}+\frac{930067953}{35395827719}a^{21}+\frac{376577839}{35395827719}a^{20}-\frac{1764797991}{35395827719}a^{19}-\frac{2308413661}{35395827719}a^{18}+\frac{1860157384}{35395827719}a^{17}+\frac{122605960}{1862938301}a^{16}-\frac{748408082}{35395827719}a^{15}+\frac{319856183}{35395827719}a^{14}+\frac{349971554}{35395827719}a^{13}-\frac{30650815}{1862938301}a^{12}+\frac{12209144223}{35395827719}a^{11}+\frac{887295919}{1862938301}a^{10}+\frac{16251295045}{35395827719}a^{9}+\frac{6924113565}{35395827719}a^{8}+\frac{9670463253}{35395827719}a^{7}+\frac{11426185426}{35395827719}a^{6}+\frac{95372290}{5056546817}a^{5}-\frac{7135920315}{35395827719}a^{4}-\frac{6120440328}{35395827719}a^{3}+\frac{4544595729}{35395827719}a^{2}-\frac{768658820}{1862938301}a-\frac{916766278}{1862938301}$, $\frac{1}{22\!\cdots\!33}a^{43}-\frac{8301503762}{22\!\cdots\!33}a^{42}+\frac{501604087}{45\!\cdots\!17}a^{41}-\frac{991117775669565}{22\!\cdots\!33}a^{40}+\frac{93\!\cdots\!37}{22\!\cdots\!33}a^{39}-\frac{22\!\cdots\!88}{22\!\cdots\!33}a^{38}+\frac{33\!\cdots\!43}{31\!\cdots\!19}a^{37}+\frac{98\!\cdots\!08}{22\!\cdots\!33}a^{36}-\frac{45\!\cdots\!29}{22\!\cdots\!33}a^{35}+\frac{43\!\cdots\!97}{31\!\cdots\!19}a^{34}-\frac{56\!\cdots\!15}{31\!\cdots\!19}a^{33}+\frac{15\!\cdots\!26}{31\!\cdots\!19}a^{32}+\frac{26\!\cdots\!79}{31\!\cdots\!19}a^{31}+\frac{53\!\cdots\!42}{31\!\cdots\!19}a^{30}-\frac{15\!\cdots\!68}{22\!\cdots\!33}a^{29}+\frac{10\!\cdots\!19}{22\!\cdots\!33}a^{28}+\frac{15\!\cdots\!43}{31\!\cdots\!19}a^{27}+\frac{85\!\cdots\!96}{22\!\cdots\!33}a^{26}+\frac{10\!\cdots\!80}{22\!\cdots\!33}a^{25}-\frac{17\!\cdots\!39}{22\!\cdots\!33}a^{24}+\frac{13\!\cdots\!15}{31\!\cdots\!19}a^{23}+\frac{12\!\cdots\!99}{22\!\cdots\!33}a^{22}+\frac{67\!\cdots\!04}{22\!\cdots\!33}a^{21}-\frac{19\!\cdots\!90}{31\!\cdots\!19}a^{20}+\frac{14\!\cdots\!56}{22\!\cdots\!33}a^{19}-\frac{80\!\cdots\!78}{22\!\cdots\!33}a^{18}-\frac{71\!\cdots\!57}{22\!\cdots\!33}a^{17}+\frac{20\!\cdots\!90}{31\!\cdots\!19}a^{16}-\frac{33\!\cdots\!32}{22\!\cdots\!33}a^{15}-\frac{94\!\cdots\!32}{22\!\cdots\!33}a^{14}+\frac{10\!\cdots\!56}{31\!\cdots\!19}a^{13}+\frac{15\!\cdots\!04}{22\!\cdots\!33}a^{12}-\frac{97\!\cdots\!85}{22\!\cdots\!33}a^{11}-\frac{15\!\cdots\!14}{22\!\cdots\!33}a^{10}-\frac{84\!\cdots\!44}{45\!\cdots\!17}a^{9}+\frac{34\!\cdots\!61}{11\!\cdots\!07}a^{8}+\frac{31\!\cdots\!05}{22\!\cdots\!33}a^{7}+\frac{72\!\cdots\!01}{31\!\cdots\!19}a^{6}+\frac{11\!\cdots\!97}{22\!\cdots\!33}a^{5}+\frac{66\!\cdots\!85}{22\!\cdots\!33}a^{4}+\frac{90\!\cdots\!98}{22\!\cdots\!33}a^{3}+\frac{30\!\cdots\!81}{31\!\cdots\!19}a^{2}-\frac{64\!\cdots\!79}{22\!\cdots\!33}a-\frac{27\!\cdots\!75}{22\!\cdots\!33}$, $\frac{1}{58\!\cdots\!33}a^{44}+\frac{29\!\cdots\!15}{58\!\cdots\!33}a^{43}-\frac{82\!\cdots\!54}{58\!\cdots\!33}a^{42}+\frac{52\!\cdots\!03}{58\!\cdots\!33}a^{41}-\frac{28\!\cdots\!18}{58\!\cdots\!33}a^{40}+\frac{34\!\cdots\!66}{58\!\cdots\!33}a^{39}+\frac{15\!\cdots\!40}{58\!\cdots\!33}a^{38}-\frac{58\!\cdots\!79}{58\!\cdots\!33}a^{37}+\frac{26\!\cdots\!65}{58\!\cdots\!33}a^{36}-\frac{99\!\cdots\!93}{58\!\cdots\!33}a^{35}+\frac{20\!\cdots\!93}{84\!\cdots\!19}a^{34}-\frac{42\!\cdots\!13}{84\!\cdots\!19}a^{33}+\frac{13\!\cdots\!19}{84\!\cdots\!19}a^{32}-\frac{30\!\cdots\!80}{84\!\cdots\!19}a^{31}-\frac{51\!\cdots\!86}{58\!\cdots\!33}a^{30}-\frac{81\!\cdots\!94}{31\!\cdots\!07}a^{29}-\frac{31\!\cdots\!75}{58\!\cdots\!33}a^{28}+\frac{28\!\cdots\!18}{58\!\cdots\!33}a^{27}-\frac{99\!\cdots\!48}{58\!\cdots\!33}a^{26}-\frac{28\!\cdots\!49}{58\!\cdots\!33}a^{25}-\frac{38\!\cdots\!71}{58\!\cdots\!33}a^{24}+\frac{26\!\cdots\!51}{58\!\cdots\!33}a^{23}+\frac{36\!\cdots\!73}{58\!\cdots\!33}a^{22}+\frac{21\!\cdots\!34}{31\!\cdots\!07}a^{21}-\frac{31\!\cdots\!07}{58\!\cdots\!33}a^{20}-\frac{19\!\cdots\!98}{58\!\cdots\!33}a^{19}+\frac{17\!\cdots\!59}{58\!\cdots\!33}a^{18}+\frac{16\!\cdots\!77}{58\!\cdots\!33}a^{17}-\frac{18\!\cdots\!46}{58\!\cdots\!33}a^{16}+\frac{92\!\cdots\!41}{84\!\cdots\!19}a^{15}+\frac{72\!\cdots\!47}{31\!\cdots\!07}a^{14}-\frac{77\!\cdots\!14}{58\!\cdots\!33}a^{13}-\frac{29\!\cdots\!91}{58\!\cdots\!33}a^{12}+\frac{21\!\cdots\!33}{58\!\cdots\!33}a^{11}-\frac{92\!\cdots\!91}{58\!\cdots\!33}a^{10}-\frac{23\!\cdots\!22}{58\!\cdots\!33}a^{9}+\frac{16\!\cdots\!45}{58\!\cdots\!33}a^{8}-\frac{22\!\cdots\!77}{58\!\cdots\!33}a^{7}-\frac{10\!\cdots\!38}{58\!\cdots\!33}a^{6}+\frac{62\!\cdots\!39}{58\!\cdots\!33}a^{5}-\frac{18\!\cdots\!63}{58\!\cdots\!33}a^{4}+\frac{98\!\cdots\!22}{77\!\cdots\!69}a^{3}-\frac{31\!\cdots\!40}{58\!\cdots\!33}a^{2}+\frac{15\!\cdots\!03}{58\!\cdots\!33}a+\frac{10\!\cdots\!69}{58\!\cdots\!33}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/7*a^12 - 2/7*a^11 + 3/7*a^10 - 1/7*a^9 - 3/7*a^8 - 3/7*a^7 + 3/7*a^6 - 2/7*a^5 + 1/7*a^4 + 1/7*a^3 - 2/7*a^2 - 3/7*a + 1/7, 1/7*a^13 - 1/7*a^11 - 2/7*a^10 + 2/7*a^9 - 2/7*a^8 - 3/7*a^7 - 3/7*a^6 - 3/7*a^5 + 3/7*a^4 + 2/7*a + 2/7, 1/7*a^14 + 3/7*a^11 - 2/7*a^10 - 3/7*a^9 + 1/7*a^8 + 1/7*a^7 + 1/7*a^5 + 1/7*a^4 + 1/7*a^3 - 1/7*a + 1/7, 1/7*a^15 - 3/7*a^11 + 2/7*a^10 - 3/7*a^9 + 3/7*a^8 + 2/7*a^7 - 1/7*a^6 - 2/7*a^4 - 3/7*a^3 - 2/7*a^2 + 3/7*a - 3/7, 1/7*a^16 + 3/7*a^11 - 1/7*a^10 - 3/7*a^7 + 2/7*a^6 - 1/7*a^5 + 1/7*a^3 - 3/7*a^2 + 2/7*a + 3/7, 1/7*a^17 - 2/7*a^11 - 2/7*a^10 + 3/7*a^9 - 1/7*a^8 - 3/7*a^7 - 3/7*a^6 - 1/7*a^5 - 2/7*a^4 + 1/7*a^3 + 1/7*a^2 - 2/7*a - 3/7, 1/7*a^18 + 1/7*a^11 + 2/7*a^10 - 3/7*a^9 - 2/7*a^8 - 2/7*a^7 - 2/7*a^6 + 1/7*a^5 + 3/7*a^4 + 3/7*a^3 + 1/7*a^2 - 2/7*a + 2/7, 1/7*a^19 - 3/7*a^11 + 1/7*a^10 - 1/7*a^9 + 1/7*a^8 + 1/7*a^7 - 2/7*a^6 - 2/7*a^5 + 2/7*a^4 - 2/7*a - 1/7, 1/7*a^20 + 2/7*a^11 + 1/7*a^10 - 2/7*a^9 - 1/7*a^8 + 3/7*a^7 + 3/7*a^5 + 3/7*a^4 + 3/7*a^3 - 1/7*a^2 - 3/7*a + 3/7, 1/7*a^21 - 2/7*a^11 - 1/7*a^10 + 1/7*a^9 + 2/7*a^8 - 1/7*a^7 - 3/7*a^6 + 1/7*a^4 - 3/7*a^3 + 1/7*a^2 + 2/7*a - 2/7, 1/7*a^22 + 2/7*a^11 - 2/7*a^7 - 1/7*a^6 - 3/7*a^5 - 1/7*a^4 + 3/7*a^3 - 2/7*a^2 - 1/7*a + 2/7, 1/7*a^23 - 3/7*a^11 + 1/7*a^10 + 2/7*a^9 - 3/7*a^8 - 2/7*a^7 - 2/7*a^6 + 3/7*a^5 + 1/7*a^4 + 3/7*a^3 + 3/7*a^2 + 1/7*a - 2/7, 1/49*a^24 + 3/49*a^23 + 3/49*a^22 + 1/49*a^18 + 1/49*a^16 - 2/49*a^15 + 1/49*a^12 - 1/7*a^11 - 1/7*a^10 + 2/49*a^9 - 17/49*a^8 - 1/7*a^7 + 1/49*a^6 + 1/7*a^5 - 1/7*a^4 - 3/7*a^3 - 2/49*a^2 + 1/49*a + 1/49, 1/931*a^25 - 2/931*a^24 + 16/931*a^23 - 43/931*a^22 + 3/133*a^21 - 4/133*a^20 - 55/931*a^19 - 26/931*a^18 + 15/931*a^17 - 2/133*a^16 - 32/931*a^15 + 8/133*a^14 + 22/931*a^13 - 40/931*a^12 - 46/133*a^11 - 327/931*a^10 - 48/931*a^9 - 384/931*a^8 + 183/931*a^7 + 345/931*a^6 + 11/133*a^5 + 53/133*a^4 + 201/931*a^3 - 2/49*a^2 + 185/931*a + 149/931, 1/931*a^26 - 1/133*a^24 + 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175473404343730385660272923211812089793515862106623283355344597007501632725435836641336684563168605966398549652938203276848347110465175225442213557894911320381043323558284128312683297455635189695059621159825139604270514968588295980660403250136780822082770991265884810034405088697200378781803824954469332873286560596963548573373357804383268788441358968077284985144959/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^18 + 163747318436627827426110236981824336043511675245304240515142018379726758399234632097592630464175615074637618315432358710043104803265461244747868775467629450021625464994394196288771586825893163413599994756212597364443463107755381894332847130219728225863539651388213863970354581119753119676212014873670160232094406362268003789137956978354453448931791590425318078548877/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^17 - 182862564048001374593344420987449553214397387340614129848890530426884778106625273169162696720714795566689558178629050657096554592026542129293117771360803436158007809629802590947984548653010813368524846394259044832684053044844679372195554541585289659944974406972721384140017426805085697126192220259049320006738516993849018856115379820507298282300129716053228866591646/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^16 + 9227988982116607477063382911712650243575315496512200463253937696797257947823342018713467146778025508299794788239877925863682209476899081256614899969699167191202014230748515008522000377942843521682246148297738441628871076749851827425262605147245684168785906901507362004424876992413985416415641680761391392063962939701527376929223460132390135194691605926880768578441/841835732577357117453065332445336714542413614429286511226559784849343871696615516481439538005175798433775450511100774182946806613764957720735479545159130024361189821865104591858103010084007445565907326024710191232709943247599930969757414160350793415721013729171150415122908311304049867052809734883733375105358608667837522039122877420216811497119199872470320010798619*a^15 + 727347658728776276848237247396482862780153352935322513091879556495015788793712108222329628537094553663659816827526135428291722408735613328415333149471250585338659658911163388024506285523675691652795911446642480072505926659001446993117516515749091702055689842312610080753750359840642834261464461842657787590696270743949415829353589475074916964166344623587086319047/310150006739026306430076701427229315884047121105526609399258868102389847467174137651056671896643715212443587030405548383190928752439721265534124042953363693185701513318722744368774793188844848366386909588051123085735242249115764041489573638023976521581426110747265942413703062059386793124719376009796506617763697930255929172308428523237772656833389426699591582925807*a^14 - 7720418766464533224209716357066349523022737125342501358569710367790069809919036661256340254724204616317825054597083987416742609749570330196550731402871493008404247247792164446131975905012887312378421859140150652572531161667364856659033312361403581380745452074483977871874735345727326165093671235674366024899539874565729120593343728137507776059350707027520025506814/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^13 - 296429884189912737343568546219020324791244936343249028810789744960493317854912253339861268908494092775736012207798152486956203972889979354937975925010752007678816568295142150477648502582132326677014869941253536901641409254191141095394408684646728336481288995293826280886513967855065518728318393855108431353768846786325693750285542821897210302030599831197991318588391/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^12 + 2179358235594042262917854526059694631290629480317044549891555541885716599589492899244232805424389265989533715805489263778097060865674116992595715058006356121600701408433430217948393962378873457984941295379794180260312977865579177378200207527812130811995584975155124436057886240899606294491230607030968867784592170860535950266036357690620577300387993913327709178317933/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^11 - 924928713717328315185856339477000037801626922355574620651847979033637324752612264408868214973314498787215158627469350812363947900210925927286714928180922146280139178173689819142668777039853943554300219485743322543389150730360837790960830923160557069481236029321999864912670870539368522656763014329650814325713530956961161233763780446379291681483652393540098651303991/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^10 - 2354238777573689092581517892976156356044673870953152376543159669173850320036520836432530882503975036166954030368187514462951739962810538378582270104232341634289350993499651406346709659399439582455832973484924832386252867880755124741484271044774516511222586362612110374980098866269288054417498537087827420980606507403842333183796147860212245602611592172155149568203322/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^9 + 1688612954059438408701889382299098126803572210875913946827490155431599084333518665747696720360756505351246303254022159590006989382284147148583134644300818420348438749863053580920710976713856866115828272859962893093722432006752168232320758589599927402112744118932179854738627974689602462269085096259147008621277206601583574889582882980703215710558299225023394858600745/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^8 - 220474996355468048567150513953810095722752070007626732608338381915075253769079743625321879105374458804344853123775350252365102284527305268960121266519326966226563839996410052026945850798383382326123968465070266012399550859871725897772062664586485966177598063444423747313513800386514861755714466310292441162653086821573284675504532487248102309270816825040075443427177/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^7 - 1036634115842069021893915524458666418816186782190990970541291840204349015642880702856383445858521227445256128578410198552751989553156764487041187344261159772134564553395701946382141351262557125891244076096171095280699532198478653618004321851069506716507504743927411786386510340963393969625072854515236405523143403937873076483719754802070623262696620117153043728076238/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^6 + 627342390652598039865441869669414549323983338780037122937820523634264353628449705709770716614584887754479223977703777720615681555158717120512939607167842329400861811116157200517482636889544440822584148041611235042667353633359352232616596843317231456997111839197832242802839676031096227052085795139710338020107074132855836054097050006545934027785083846165703534195239/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^5 - 180751943164323486396045043826332621184779758352849573640934001539338157216647643243810763690730855644268417956984681311710189369544649212343408094056101388488312802108252671529225184589593715120223221608212046001505025587710215268904030510586880922192490926097872939142539897723830951921135167320590448937133045161963866934127358066939847239971338248018938424870063/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^4 + 98945451399726343904354635192027058430534037366541454261629995626262223919381597475758128177465664161069312618070132869685118420388829441954933301237969232091316095354760699350473271016024838119194551984774165648496447695998381601773329570817799684236316817666329379133046711597389734583058142249866025038816572026045251653069884310577113441913084865509641233122/7784478372577938998905491845597565392069874902252319126269377138633298681474648104848185952491718083271371404990363829961199004354497627536523588924853249895017607335608628986798838930763609139975364969845404674542892473887978225612023644811698221809837643466576027616724383327778532456234700322570850232149947504194006148314214190147315297859754820485194504723369*a^3 - 310626353606584548940926708814816970940292338729714331445931455475113700489988899544994275143023659066664077988724254796983125067592395360288780886329047078872121236311959013867130983496997720577685243415075639700546038807068238197994764815769145452433476934634877302180157798091834264942334238429470983809237568308215822096101106664057483673896091816503363272970440/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a^2 + 1519419794906361057263628157295499316805791391350039699876385288053427492545790626955444437999725663920908113135342219315650188423054926619468199211430583344005024765918919119074391440656729241554979910483168502923934135252486613059267883735116484761052693899493812592654319465476719696837172955099965740225705349059353279615971707076880732964477123122543230084733503/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333*a + 1050808077532511089959026810276429464117026006425911347582083261627352062878218910827681053374084561612237088997447657889456665206754937658948342945269853370551768063913824764093034172093283194092270072220035749121146882836353882687993644053012713030513576689734653514017575257024736900148293734493835683833114324055833661620580622333806188597126320933080490502293369/5892850128041499822171457327117357001796895301005005578585918493945407101876308615370076766036230589036428153577705419280627646296354704045148356816113910170528328753055732143006721070588052118961351282172971338628969602733199516788301899122455553910047096104198052905860358179128349069369668144186133625737510260674862654273860141941517680479834399107292240075590333

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:	$44$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651)

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^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651, 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^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651);

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^45 - 405*x^43 - 45*x^42 + 74565*x^41 + 13986*x^40 - 8290800*x^39 - 1937610*x^38 + 623822085*x^37 + 158343200*x^36 - 33730092612*x^35 - 8501973165*x^34 + 1358280567105*x^33 + 315527932755*x^32 - 41656181730075*x^31 - 8257524539733*x^30 + 986564158362990*x^29 + 151291718024085*x^28 - 18188682238507875*x^27 - 1837307213515305*x^26 + 261951082234160256*x^25 + 11633106453685350*x^24 - 2946259043332552305*x^23 + 39719938240377450*x^22 + 25786532715466797195*x^21 - 1805631856520318496*x^20 - 174433501954113158925*x^19 + 21474396909776230700*x^18 + 902723065221863783160*x^17 - 154718694923268688020*x^16 - 3523794539226841295061*x^15 + 755660668535950676940*x^14 + 10175405955078415341510*x^13 - 2572756413109899568440*x^12 - 21151381993350223213305*x^11 + 6092956812785525217882*x^10 + 30403883242511231332790*x^9 - 9799931675663506270395*x^8 - 28334864017939625412120*x^7 + 10152119383093656646320*x^6 + 15176238217723477388907*x^5 - 6060909987353410350210*x^4 - 3421782109383146916285*x^3 + 1568220560944780065255*x^2 - 73641665210792649300*x - 1914182976156359651);

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

$C_{45}$ (as 45T1):

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 45

The 45 conjugacy class representatives for $C_{45}$

Character table for $C_{45}$ is not computed

Intermediate fields

$\Q(\zeta_{9})^+$, 5.5.390625.1, 9.9.3691950281939241.2, 15.15.207828545629978179931640625.1

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	$45$	R	R	R	$45$	$45$	${\href{/padicField/17.5.0.1}{5} }^{9}$	${\href{/padicField/19.5.0.1}{5} }^{9}$	$45$	$45$	$45$	$15^{3}$	$45$	${\href{/padicField/43.9.0.1}{9} }^{5}$	$45$	$15^{3}$	$45$

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		Deg $45$	$9$	$5$	$110$
$5$ 5		Deg $45$	$5$	$9$	$72$
$7$ 7	7.9.6.3	$x^{9} - 42 x^{6} - 1372$	$3$	$3$	$6$	$C_9$	$[\ ]_{3}^{3}$
	7.9.6.3	$x^{9} - 42 x^{6} - 1372$	$3$	$3$	$6$	$C_9$	$[\ ]_{3}^{3}$
	7.9.6.3	$x^{9} - 42 x^{6} - 1372$	$3$	$3$	$6$	$C_9$	$[\ ]_{3}^{3}$
	7.9.6.3	$x^{9} - 42 x^{6} - 1372$	$3$	$3$	$6$	$C_9$	$[\ ]_{3}^{3}$
	7.9.6.3	$x^{9} - 42 x^{6} - 1372$	$3$	$3$	$6$	$C_9$	$[\ ]_{3}^{3}$

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