LMFDB - Number field 28.28.1159427335739550761098088865697701264851767903254485069.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911)

gp: K = bnfinit(y^28 - y^27 - 86*y^26 + 86*y^25 + 3307*y^24 - 3307*y^23 - 74993*y^22 + 74993*y^21 + 1113601*y^20 - 1113601*y^19 - 11366636*y^18 + 11366636*y^17 + 81421213*y^16 - 81421213*y^15 - 410233883*y^14 + 410233883*y^13 + 1433472727*y^12 - 1433472727*y^11 - 3360164459*y^10 + 3360164459*y^9 + 4965626443*y^8 - 4965626443*y^7 - 4117054541*y^6 + 4117054541*y^5 + 1492836655*y^4 - 1492836655*y^3 - 125401190*y^2 + 125401190*y + 13304911, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911)

$ x^{28} - x^{27} - 86 x^{26} + 86 x^{25} + 3307 x^{24} - 3307 x^{23} - 74993 x^{22} + 74993 x^{21} + \cdots + 13304911 $ x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$28$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[28, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$1159427335739550761098088865697701264851767903254485069$ 1159427335739550761098088865697701264851767903254485069 $\medspace = 11^{14}\cdot 29^{27}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$85.28$	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:	$11^{1/2}29^{27/28}\approx 85.2836620539706$
Ramified primes:	$11$, $29$ 11, 29	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{29}) $
$\card{ \Gal(K/\Q) }$:	$28$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$319=11\cdot 29$
Dirichlet character group:	$\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(131,·)$, $\chi_{319}(100,·)$, $\chi_{319}(263,·)$, $\chi_{319}(265,·)$, $\chi_{319}(10,·)$, $\chi_{319}(76,·)$, $\chi_{319}(98,·)$, $\chi_{319}(142,·)$, $\chi_{319}(144,·)$, $\chi_{319}(210,·)$, $\chi_{319}(67,·)$, $\chi_{319}(21,·)$, $\chi_{319}(23,·)$, $\chi_{319}(153,·)$, $\chi_{319}(122,·)$, $\chi_{319}(199,·)$, $\chi_{319}(32,·)$, $\chi_{319}(34,·)$, $\chi_{319}(164,·)$, $\chi_{319}(230,·)$, $\chi_{319}(43,·)$, $\chi_{319}(78,·)$, $\chi_{319}(45,·)$, $\chi_{319}(111,·)$, $\chi_{319}(307,·)$, $\chi_{319}(186,·)$, $\chi_{319}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2977199}a^{15}+\frac{742231}{2977199}a^{14}-\frac{45}{2977199}a^{13}-\frac{1401712}{2977199}a^{12}+\frac{810}{2977199}a^{11}-\frac{689344}{2977199}a^{10}-\frac{7425}{2977199}a^{9}+\frac{1309616}{2977199}a^{8}+\frac{36450}{2977199}a^{7}-\frac{141429}{2977199}a^{6}-\frac{91854}{2977199}a^{5}+\frac{282858}{2977199}a^{4}+\frac{102060}{2977199}a^{3}+\frac{1276456}{2977199}a^{2}-\frac{32805}{2977199}a-\frac{1371484}{2977199}$, $\frac{1}{2977199}a^{16}-\frac{48}{2977199}a^{14}-\frac{750506}{2977199}a^{13}+\frac{936}{2977199}a^{12}-\frac{502256}{2977199}a^{11}-\frac{9504}{2977199}a^{10}-\frac{1397757}{2977199}a^{9}+\frac{53460}{2977199}a^{8}-\frac{654066}{2977199}a^{7}-\frac{163296}{2977199}a^{6}-\frac{687968}{2977199}a^{5}+\frac{244944}{2977199}a^{4}+\frac{1031952}{2977199}a^{3}-\frac{139968}{2977199}a^{2}-\frac{16951}{2977199}a+\frac{13122}{2977199}$, $\frac{1}{2977199}a^{17}-\frac{849806}{2977199}a^{14}-\frac{1224}{2977199}a^{13}+\frac{691145}{2977199}a^{12}+\frac{29376}{2977199}a^{11}+\frac{1240119}{2977199}a^{10}-\frac{302940}{2977199}a^{9}-\frac{313677}{2977199}a^{8}-\frac{1390895}{2977199}a^{7}+\frac{1455037}{2977199}a^{6}-\frac{1186849}{2977199}a^{5}-\frac{276859}{2977199}a^{4}-\frac{1195486}{2977199}a^{3}-\frac{1268242}{2977199}a^{2}+\frac{1415681}{2977199}a-\frac{332854}{2977199}$, $\frac{1}{2977199}a^{18}-\frac{1377}{2977199}a^{14}+\frac{1153462}{2977199}a^{13}+\frac{35802}{2977199}a^{12}-\frac{1127189}{2977199}a^{11}-\frac{408969}{2977199}a^{10}-\frac{1438546}{2977199}a^{9}-\frac{523385}{2977199}a^{8}-\frac{871858}{2977199}a^{7}+\frac{1124007}{2977199}a^{6}+\frac{823398}{2977199}a^{5}+\frac{137200}{2977199}a^{4}+\frac{1148049}{2977199}a^{3}-\frac{1072433}{2977199}a^{2}+\frac{272752}{2977199}a+\frac{669222}{2977199}$, $\frac{1}{2977199}a^{19}-\frac{950907}{2977199}a^{14}-\frac{26163}{2977199}a^{13}+\frac{917538}{2977199}a^{12}+\frac{706401}{2977199}a^{11}-\frac{938753}{2977199}a^{10}+\frac{1161186}{2977199}a^{9}+\frac{1263979}{2977199}a^{8}+\frac{703274}{2977199}a^{7}-\frac{406400}{2977199}a^{6}-\frac{1303400}{2977199}a^{5}+\frac{630446}{2977199}a^{4}-\frac{464166}{2977199}a^{3}+\frac{1405254}{2977199}a^{2}+\frac{154722}{2977199}a-\frac{989302}{2977199}$, $\frac{1}{2977199}a^{20}-\frac{30780}{2977199}a^{14}-\frac{192491}{2977199}a^{13}+\frac{900315}{2977199}a^{12}+\frac{1178575}{2977199}a^{11}+\frac{938804}{2977199}a^{10}-\frac{281667}{2977199}a^{9}+\frac{86873}{2977199}a^{8}-\frac{397008}{2977199}a^{7}-\frac{1096275}{2977199}a^{6}+\frac{1083130}{2977199}a^{5}-\frac{878416}{2977199}a^{4}+\frac{240672}{2977199}a^{3}-\frac{1023190}{2977199}a^{2}-\frac{402315}{2977199}a-\frac{645635}{2977199}$, $\frac{1}{2977199}a^{21}-\frac{1347437}{2977199}a^{14}-\frac{484785}{2977199}a^{13}-\frac{926076}{2977199}a^{12}-\frac{924187}{2977199}a^{11}+\frac{207286}{2977199}a^{10}+\frac{789696}{2977199}a^{9}+\frac{1286211}{2977199}a^{8}+\frac{1407901}{2977199}a^{7}+\frac{563448}{2977199}a^{6}+\frac{194514}{2977199}a^{5}+\frac{1280036}{2977199}a^{4}-\frac{561335}{2977199}a^{3}-\frac{1181838}{2977199}a^{2}-\frac{1113074}{2977199}a-\frac{572899}{2977199}$, $\frac{1}{2977199}a^{22}-\frac{592515}{2977199}a^{14}+\frac{960438}{2977199}a^{13}+\frac{623274}{2977199}a^{12}-\frac{1000777}{2977199}a^{11}+\frac{562781}{2977199}a^{10}-\frac{44874}{2977199}a^{9}-\frac{1065993}{2977199}a^{8}-\frac{209805}{2977199}a^{7}+\frac{1057832}{2977199}a^{6}-\frac{1058533}{2977199}a^{5}+\frac{689228}{2977199}a^{4}+\frac{1416572}{2977199}a^{3}+\frac{181903}{2977199}a^{2}-\frac{770131}{2977199}a+\frac{813578}{2977199}$, $\frac{1}{2977199}a^{23}+\frac{1056720}{2977199}a^{14}+\frac{754890}{2977199}a^{13}+\frac{909777}{2977199}a^{12}+\frac{1170892}{2977199}a^{11}+\frac{1180174}{2977199}a^{10}-\frac{189746}{2977199}a^{9}-\frac{1301328}{2977199}a^{8}-\frac{1349163}{2977199}a^{7}-\frac{10885}{50461}a^{6}-\frac{985862}{2977199}a^{5}+\frac{583936}{2977199}a^{4}-\frac{603285}{2977199}a^{3}-\frac{145654}{2977199}a^{2}-\frac{1485925}{2977199}a-\frac{1352409}{2977199}$, $\frac{1}{2977199}a^{24}-\frac{1396875}{2977199}a^{14}+\frac{826993}{2977199}a^{13}-\frac{748147}{2977199}a^{12}-\frac{306913}{2977199}a^{11}+\frac{213808}{2977199}a^{10}-\frac{74693}{2977199}a^{9}-\frac{1403115}{2977199}a^{8}+\frac{914447}{2977199}a^{7}+\frac{431616}{2977199}a^{6}-\frac{1076181}{2977199}a^{5}-\frac{461042}{2977199}a^{4}+\frac{44921}{2977199}a^{3}+\frac{640292}{2977199}a^{2}+\frac{819234}{2977199}a+\frac{894071}{2977199}$, $\frac{1}{2977199}a^{25}+\frac{1157766}{2977199}a^{14}-\frac{1086343}{2977199}a^{13}+\frac{686616}{2977199}a^{12}+\frac{346938}{2977199}a^{11}-\frac{93327}{2977199}a^{10}-\frac{638674}{2977199}a^{9}+\frac{1066907}{2977199}a^{8}+\frac{468068}{2977199}a^{7}+\frac{1260686}{2977199}a^{6}-\frac{671989}{2977199}a^{5}+\frac{1325585}{2977199}a^{4}-\frac{448522}{2977199}a^{3}-\frac{141264}{2977199}a^{2}+\frac{1456704}{2977199}a+\frac{117612}{2977199}$, $\frac{1}{2977199}a^{26}-\frac{1114526}{2977199}a^{14}-\frac{803496}{2977199}a^{13}-\frac{1446575}{2977199}a^{12}-\frac{66102}{2977199}a^{11}+\frac{670900}{2977199}a^{10}-\frac{671255}{2977199}a^{9}-\frac{503068}{2977199}a^{8}-\frac{491388}{2977199}a^{7}+\frac{1025023}{2977199}a^{6}+\frac{1215469}{2977199}a^{5}-\frac{865347}{2977199}a^{4}+\frac{311887}{2977199}a^{3}+\frac{1025023}{2977199}a^{2}+\frac{503599}{2977199}a+\frac{1207283}{2977199}$, $\frac{1}{2977199}a^{27}-\frac{638533}{2977199}a^{14}-\frac{987862}{2977199}a^{13}-\frac{1040150}{2977199}a^{12}+\frac{1345663}{2977199}a^{11}+\frac{514542}{2977199}a^{10}+\frac{754602}{2977199}a^{9}-\frac{991112}{2977199}a^{8}-\frac{1359831}{2977199}a^{7}-\frac{258329}{2977199}a^{6}-\frac{571737}{2977199}a^{5}+\frac{282284}{2977199}a^{4}-\frac{293610}{2977199}a^{3}+\frac{1270101}{2977199}a^{2}-\frac{814427}{2977199}a-\frac{1066004}{2977199}$ 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, 1/2977199*a^15 + 742231/2977199*a^14 - 45/2977199*a^13 - 1401712/2977199*a^12 + 810/2977199*a^11 - 689344/2977199*a^10 - 7425/2977199*a^9 + 1309616/2977199*a^8 + 36450/2977199*a^7 - 141429/2977199*a^6 - 91854/2977199*a^5 + 282858/2977199*a^4 + 102060/2977199*a^3 + 1276456/2977199*a^2 - 32805/2977199*a - 1371484/2977199, 1/2977199*a^16 - 48/2977199*a^14 - 750506/2977199*a^13 + 936/2977199*a^12 - 502256/2977199*a^11 - 9504/2977199*a^10 - 1397757/2977199*a^9 + 53460/2977199*a^8 - 654066/2977199*a^7 - 163296/2977199*a^6 - 687968/2977199*a^5 + 244944/2977199*a^4 + 1031952/2977199*a^3 - 139968/2977199*a^2 - 16951/2977199*a + 13122/2977199, 1/2977199*a^17 - 849806/2977199*a^14 - 1224/2977199*a^13 + 691145/2977199*a^12 + 29376/2977199*a^11 + 1240119/2977199*a^10 - 302940/2977199*a^9 - 313677/2977199*a^8 - 1390895/2977199*a^7 + 1455037/2977199*a^6 - 1186849/2977199*a^5 - 276859/2977199*a^4 - 1195486/2977199*a^3 - 1268242/2977199*a^2 + 1415681/2977199*a - 332854/2977199, 1/2977199*a^18 - 1377/2977199*a^14 + 1153462/2977199*a^13 + 35802/2977199*a^12 - 1127189/2977199*a^11 - 408969/2977199*a^10 - 1438546/2977199*a^9 - 523385/2977199*a^8 - 871858/2977199*a^7 + 1124007/2977199*a^6 + 823398/2977199*a^5 + 137200/2977199*a^4 + 1148049/2977199*a^3 - 1072433/2977199*a^2 + 272752/2977199*a + 669222/2977199, 1/2977199*a^19 - 950907/2977199*a^14 - 26163/2977199*a^13 + 917538/2977199*a^12 + 706401/2977199*a^11 - 938753/2977199*a^10 + 1161186/2977199*a^9 + 1263979/2977199*a^8 + 703274/2977199*a^7 - 406400/2977199*a^6 - 1303400/2977199*a^5 + 630446/2977199*a^4 - 464166/2977199*a^3 + 1405254/2977199*a^2 + 154722/2977199*a - 989302/2977199, 1/2977199*a^20 - 30780/2977199*a^14 - 192491/2977199*a^13 + 900315/2977199*a^12 + 1178575/2977199*a^11 + 938804/2977199*a^10 - 281667/2977199*a^9 + 86873/2977199*a^8 - 397008/2977199*a^7 - 1096275/2977199*a^6 + 1083130/2977199*a^5 - 878416/2977199*a^4 + 240672/2977199*a^3 - 1023190/2977199*a^2 - 402315/2977199*a - 645635/2977199, 1/2977199*a^21 - 1347437/2977199*a^14 - 484785/2977199*a^13 - 926076/2977199*a^12 - 924187/2977199*a^11 + 207286/2977199*a^10 + 789696/2977199*a^9 + 1286211/2977199*a^8 + 1407901/2977199*a^7 + 563448/2977199*a^6 + 194514/2977199*a^5 + 1280036/2977199*a^4 - 561335/2977199*a^3 - 1181838/2977199*a^2 - 1113074/2977199*a - 572899/2977199, 1/2977199*a^22 - 592515/2977199*a^14 + 960438/2977199*a^13 + 623274/2977199*a^12 - 1000777/2977199*a^11 + 562781/2977199*a^10 - 44874/2977199*a^9 - 1065993/2977199*a^8 - 209805/2977199*a^7 + 1057832/2977199*a^6 - 1058533/2977199*a^5 + 689228/2977199*a^4 + 1416572/2977199*a^3 + 181903/2977199*a^2 - 770131/2977199*a + 813578/2977199, 1/2977199*a^23 + 1056720/2977199*a^14 + 754890/2977199*a^13 + 909777/2977199*a^12 + 1170892/2977199*a^11 + 1180174/2977199*a^10 - 189746/2977199*a^9 - 1301328/2977199*a^8 - 1349163/2977199*a^7 - 10885/50461*a^6 - 985862/2977199*a^5 + 583936/2977199*a^4 - 603285/2977199*a^3 - 145654/2977199*a^2 - 1485925/2977199*a - 1352409/2977199, 1/2977199*a^24 - 1396875/2977199*a^14 + 826993/2977199*a^13 - 748147/2977199*a^12 - 306913/2977199*a^11 + 213808/2977199*a^10 - 74693/2977199*a^9 - 1403115/2977199*a^8 + 914447/2977199*a^7 + 431616/2977199*a^6 - 1076181/2977199*a^5 - 461042/2977199*a^4 + 44921/2977199*a^3 + 640292/2977199*a^2 + 819234/2977199*a + 894071/2977199, 1/2977199*a^25 + 1157766/2977199*a^14 - 1086343/2977199*a^13 + 686616/2977199*a^12 + 346938/2977199*a^11 - 93327/2977199*a^10 - 638674/2977199*a^9 + 1066907/2977199*a^8 + 468068/2977199*a^7 + 1260686/2977199*a^6 - 671989/2977199*a^5 + 1325585/2977199*a^4 - 448522/2977199*a^3 - 141264/2977199*a^2 + 1456704/2977199*a + 117612/2977199, 1/2977199*a^26 - 1114526/2977199*a^14 - 803496/2977199*a^13 - 1446575/2977199*a^12 - 66102/2977199*a^11 + 670900/2977199*a^10 - 671255/2977199*a^9 - 503068/2977199*a^8 - 491388/2977199*a^7 + 1025023/2977199*a^6 + 1215469/2977199*a^5 - 865347/2977199*a^4 + 311887/2977199*a^3 + 1025023/2977199*a^2 + 503599/2977199*a + 1207283/2977199, 1/2977199*a^27 - 638533/2977199*a^14 - 987862/2977199*a^13 - 1040150/2977199*a^12 + 1345663/2977199*a^11 + 514542/2977199*a^10 + 754602/2977199*a^9 - 991112/2977199*a^8 - 1359831/2977199*a^7 - 258329/2977199*a^6 - 571737/2977199*a^5 + 282284/2977199*a^4 - 293610/2977199*a^3 + 1270101/2977199*a^2 - 814427/2977199*a - 1066004/2977199

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

Trivial group, which has order $1$ (assuming GRH)

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:	$27$	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:	$\frac{2}{2977199}a^{26}-\frac{156}{2977199}a^{24}+\frac{5382}{2977199}a^{22}-\frac{108108}{2977199}a^{20}+\frac{1400490}{2977199}a^{18}-\frac{12244284}{2977199}a^{16}+\frac{73465704}{2977199}a^{14}-\frac{301596048}{2977199}a^{12}+\frac{829389132}{2977199}a^{10}-\frac{1463627880}{2977199}a^{8}+\frac{1536809274}{2977199}a^{6}-\frac{838259604}{2977199}a^{4}+\frac{933841}{2977199}a^{3}+\frac{179627058}{2977199}a^{2}-\frac{8404569}{2977199}a-\frac{6377292}{2977199}$, $\frac{16}{2977199}a^{23}-\frac{1104}{2977199}a^{21}+\frac{33120}{2977199}a^{19}-\frac{566352}{2977199}a^{17}+\frac{6080832}{2977199}a^{15}-\frac{42565824}{2977199}a^{13}+\frac{195302016}{2977199}a^{11}-\frac{575443440}{2977199}a^{9}+\frac{1035798192}{2977199}a^{7}+\frac{166427}{2977199}a^{6}-\frac{1035798192}{2977199}a^{5}-\frac{2995686}{2977199}a^{4}+\frac{478060704}{2977199}a^{3}+\frac{13480587}{2977199}a^{2}-\frac{65190096}{2977199}a-\frac{8987058}{2977199}$, $\frac{160}{2977199}a^{17}-\frac{8160}{2977199}a^{15}+\frac{171360}{2977199}a^{13}-\frac{1801}{2977199}a^{12}-\frac{1909440}{2977199}a^{11}+\frac{64836}{2977199}a^{10}+\frac{12117600}{2977199}a^{9}-\frac{875286}{2977199}a^{8}-\frac{43623360}{2977199}a^{7}+\frac{5446224}{2977199}a^{6}+\frac{83280960}{2977199}a^{5}-\frac{15317505}{2977199}a^{4}-\frac{71383680}{2977199}a^{3}+\frac{15755148}{2977199}a^{2}+\frac{17845920}{2977199}a-\frac{2625858}{2977199}$, $\frac{1}{2977199}a^{24}-\frac{72}{2977199}a^{22}+\frac{2268}{2977199}a^{20}-\frac{41040}{2977199}a^{18}+\frac{470934}{2977199}a^{16}-\frac{3569184}{2977199}a^{14}+\frac{18044208}{2977199}a^{12}-\frac{60046272}{2977199}a^{10}+\frac{126660105}{2977199}a^{8}-\frac{157621464}{2977199}a^{6}+\frac{217280}{2977199}a^{5}+\frac{101328084}{2977199}a^{4}-\frac{3259200}{2977199}a^{3}-\frac{25509168}{2977199}a^{2}+\frac{9777600}{2977199}a+\frac{1062882}{2977199}$, $\frac{16}{2977199}a^{23}-\frac{1104}{2977199}a^{21}+\frac{33120}{2977199}a^{19}-\frac{566352}{2977199}a^{17}+\frac{6080832}{2977199}a^{15}-\frac{42565824}{2977199}a^{13}+\frac{195302016}{2977199}a^{11}-\frac{575443440}{2977199}a^{9}+\frac{1035798192}{2977199}a^{7}+\frac{166427}{2977199}a^{6}-\frac{1035798192}{2977199}a^{5}-\frac{2995686}{2977199}a^{4}+\frac{478060704}{2977199}a^{3}+\frac{13480587}{2977199}a^{2}-\frac{65190096}{2977199}a-\frac{6009859}{2977199}$, $\frac{160}{2977199}a^{17}-\frac{8160}{2977199}a^{15}+\frac{171360}{2977199}a^{13}-\frac{1801}{2977199}a^{12}-\frac{1909440}{2977199}a^{11}+\frac{64836}{2977199}a^{10}+\frac{12117600}{2977199}a^{9}-\frac{875286}{2977199}a^{8}-\frac{43623360}{2977199}a^{7}+\frac{5446224}{2977199}a^{6}+\frac{83280960}{2977199}a^{5}-\frac{15317505}{2977199}a^{4}-\frac{71383680}{2977199}a^{3}+\frac{15755148}{2977199}a^{2}+\frac{17845920}{2977199}a+\frac{351341}{2977199}$, $\frac{1}{2977199}a^{27}+\frac{2}{2977199}a^{26}-\frac{86}{2977199}a^{25}-\frac{157}{2977199}a^{24}+\frac{3291}{2977199}a^{23}+\frac{5441}{2977199}a^{22}-\frac{73889}{2977199}a^{21}-\frac{109444}{2977199}a^{20}+\frac{1080481}{2977199}a^{19}+\frac{23943}{50461}a^{18}-\frac{10800284}{2977199}a^{17}-\frac{12207577}{2977199}a^{16}+\frac{75340381}{2977199}a^{15}+\frac{71468798}{2977199}a^{14}-\frac{367668059}{2977199}a^{13}-\frac{280117615}{2977199}a^{12}+\frac{1238180775}{2977199}a^{11}+\frac{706974062}{2977199}a^{10}-\frac{2785053131}{2977199}a^{9}-\frac{1055834689}{2977199}a^{8}+\frac{3933813595}{2977199}a^{7}+\frac{761553348}{2977199}a^{6}-\frac{3102179405}{2977199}a^{5}-\frac{95989324}{2977199}a^{4}+\frac{1059611071}{2977199}a^{3}-\frac{66112132}{2977199}a^{2}-\frac{84398481}{2977199}a-\frac{4711513}{2977199}$, $\frac{599}{2977199}a^{16}-\frac{28752}{2977199}a^{14}+\frac{3955}{2977199}a^{13}+\frac{560664}{2977199}a^{12}-\frac{154245}{2977199}a^{11}-\frac{5692896}{2977199}a^{10}+\frac{2313675}{2977199}a^{9}+\frac{32022540}{2977199}a^{8}-\frac{16658460}{2977199}a^{7}-\frac{97814304}{2977199}a^{6}+\frac{58304610}{2977199}a^{5}+\frac{146721456}{2977199}a^{4}-\frac{87456915}{2977199}a^{3}-\frac{83840832}{2977199}a^{2}+\frac{37481535}{2977199}a+\frac{7860078}{2977199}$, $\frac{1}{2977199}a^{24}-\frac{72}{2977199}a^{22}+\frac{35}{2977199}a^{21}+\frac{2268}{2977199}a^{20}-\frac{2205}{2977199}a^{19}-\frac{41040}{2977199}a^{18}+\frac{59535}{2977199}a^{17}+\frac{470934}{2977199}a^{16}-\frac{899640}{2977199}a^{15}-\frac{3569184}{2977199}a^{14}+\frac{8334900}{2977199}a^{13}+\frac{18044208}{2977199}a^{12}-\frac{48759165}{2977199}a^{11}-\frac{60046272}{2977199}a^{10}+\frac{178783605}{2977199}a^{9}+\frac{126721231}{2977199}a^{8}-\frac{394053660}{2977199}a^{7}-\frac{159088488}{2977199}a^{6}+\frac{477628445}{2977199}a^{5}+\frac{112330764}{2977199}a^{4}-\frac{268487625}{2977199}a^{3}-\frac{51915600}{2977199}a^{2}+\frac{53178615}{2977199}a+\frac{10965294}{2977199}$, $\frac{1}{2977199}a^{24}-\frac{1}{50461}a^{22}+\frac{1410}{2977199}a^{20}-\frac{31}{2977199}a^{19}-\frac{16587}{2977199}a^{18}+\frac{1607}{2977199}a^{17}+\frac{77112}{2977199}a^{16}-\frac{34248}{2977199}a^{15}+\frac{369036}{2977199}a^{14}+\frac{385245}{2977199}a^{13}-\frac{7251263}{2977199}a^{12}-\frac{2432079}{2977199}a^{11}+\frac{44224672}{2977199}a^{10}+\frac{8349561}{2977199}a^{9}-\frac{140332377}{2977199}a^{8}-\frac{13071883}{2977199}a^{7}+\frac{234557874}{2977199}a^{6}+\frac{2309729}{2977199}a^{5}-\frac{172087551}{2977199}a^{4}+\frac{8022219}{2977199}a^{3}+\frac{20299086}{2977199}a^{2}+\frac{6728706}{2977199}a+\frac{3622681}{2977199}$, $\frac{16}{2977199}a^{23}-\frac{1104}{2977199}a^{21}+\frac{33120}{2977199}a^{19}-\frac{566192}{2977199}a^{17}+\frac{6072672}{2977199}a^{15}-\frac{42394464}{2977199}a^{13}-\frac{1801}{2977199}a^{12}+\frac{193392576}{2977199}a^{11}+\frac{64836}{2977199}a^{10}-\frac{563325840}{2977199}a^{9}-\frac{875286}{2977199}a^{8}+\frac{992174832}{2977199}a^{7}+\frac{5612651}{2977199}a^{6}-\frac{952517232}{2977199}a^{5}-\frac{18313191}{2977199}a^{4}+\frac{406677024}{2977199}a^{3}+\frac{29235735}{2977199}a^{2}-\frac{47344176}{2977199}a-\frac{8635717}{2977199}$, $\frac{1}{2977199}a^{27}+\frac{2}{2977199}a^{26}-\frac{81}{2977199}a^{25}-\frac{156}{2977199}a^{24}+\frac{2916}{2977199}a^{23}+\frac{5382}{2977199}a^{22}-\frac{61514}{2977199}a^{21}-\frac{108034}{2977199}a^{20}+\frac{844200}{2977199}a^{19}+\frac{1396050}{2977199}a^{18}-\frac{7913052}{2977199}a^{17}-\frac{12131064}{2977199}a^{16}+\frac{51759433}{2977199}a^{15}+\frac{71866586}{2977199}a^{14}-\frac{237160269}{2977199}a^{13}-\frac{287929542}{2977199}a^{12}+\frac{752575941}{2977199}a^{11}+\frac{756891630}{2977199}a^{10}-\frac{1606238495}{2977199}a^{9}-\frac{1228189606}{2977199}a^{8}+\frac{2178232047}{2977199}a^{7}+\frac{1093925526}{2977199}a^{6}-\frac{1680473061}{2977199}a^{5}-\frac{414516330}{2977199}a^{4}+\frac{579362455}{2977199}a^{3}+\frac{34643774}{2977199}a^{2}-\frac{48721185}{2977199}a-\frac{3872892}{2977199}$, $\frac{5}{2977199}a^{25}-\frac{375}{2977199}a^{23}+\frac{12375}{2977199}a^{21}-\frac{236250}{2977199}a^{19}+\frac{2885625}{2977199}a^{17}-\frac{23546700}{2977199}a^{15}+\frac{130126500}{2977199}a^{13}-\frac{483327000}{2977199}a^{11}+\frac{1172778750}{2977199}a^{9}-\frac{1759168125}{2977199}a^{7}+\frac{1477701225}{2977199}a^{5}+\frac{282001}{2977199}a^{4}-\frac{575727750}{2977199}a^{3}-\frac{3384012}{2977199}a^{2}+\frac{66430125}{2977199}a+\frac{2098819}{2977199}$, $\frac{1}{2977199}a^{27}-\frac{81}{2977199}a^{25}+\frac{2916}{2977199}a^{23}-\frac{61479}{2977199}a^{21}-\frac{43}{2977199}a^{20}+\frac{841995}{2977199}a^{19}+\frac{2420}{2977199}a^{18}-\frac{7853517}{2977199}a^{17}-\frac{57150}{2977199}a^{16}+\frac{50860872}{2977199}a^{15}+\frac{734400}{2977199}a^{14}-\frac{228873924}{2977199}a^{13}-\frac{5565105}{2977199}a^{12}+\frac{704696169}{2977199}a^{11}+\frac{25158276}{2977199}a^{10}-\frac{1435627813}{2977199}a^{9}-\frac{65194470}{2977199}a^{8}+\frac{1825189848}{2977199}a^{7}+\frac{86605200}{2977199}a^{6}-\frac{1309286106}{2977199}a^{5}-\frac{43794675}{2977199}a^{4}+\frac{433662984}{2977199}a^{3}-\frac{481498}{2977199}a^{2}-\frac{34397136}{2977199}a-\frac{4207024}{2977199}$, $\frac{4}{2977199}a^{25}-\frac{300}{2977199}a^{23}+\frac{9939}{2977199}a^{21}-\frac{191457}{2977199}a^{19}+\frac{2374999}{2977199}a^{17}-\frac{19847976}{2977199}a^{15}+\frac{113560020}{2977199}a^{13}-\frac{1801}{2977199}a^{12}-\frac{442902681}{2977199}a^{11}+\frac{64836}{2977199}a^{10}+\frac{1149556617}{2977199}a^{9}-\frac{892237}{2977199}a^{8}-\frac{1890046224}{2977199}a^{7}+\frac{5853048}{2977199}a^{6}+\frac{1797414381}{2977199}a^{5}-\frac{18738524}{2977199}a^{4}-\frac{827506125}{2977199}a^{3}+\frac{27516048}{2977199}a^{2}+\frac{119351151}{2977199}a-\frac{17983420}{2977199}$, $\frac{15}{2977199}a^{24}-\frac{1093}{2977199}a^{22}+\frac{34878}{2977199}a^{20}-\frac{640053}{2977199}a^{18}-\frac{919}{2977199}a^{17}+\frac{7457832}{2977199}a^{16}+\frac{46869}{2977199}a^{15}-\frac{57475980}{2977199}a^{14}-\frac{984249}{2977199}a^{13}+\frac{295952129}{2977199}a^{12}+\frac{10967346}{2977199}a^{11}-\frac{1004747859}{2977199}a^{10}-\frac{69600465}{2977199}a^{9}+\frac{2164217535}{2977199}a^{8}+\frac{250578625}{2977199}a^{7}-\frac{2741832315}{2977199}a^{6}-\frac{478418984}{2977199}a^{5}+\frac{1759258035}{2977199}a^{4}+\frac{407915823}{2977199}a^{3}-\frac{403236873}{2977199}a^{2}-\frac{93016197}{2977199}a+\frac{2547200}{2977199}$, $\frac{1}{2977199}a^{24}-\frac{72}{2977199}a^{22}+\frac{4}{2977199}a^{21}+\frac{2268}{2977199}a^{20}-\frac{252}{2977199}a^{19}-\frac{41040}{2977199}a^{18}+\frac{6804}{2977199}a^{17}+\frac{471414}{2977199}a^{16}-\frac{102816}{2977199}a^{15}-\frac{3592224}{2977199}a^{14}+\frac{950759}{2977199}a^{13}+\frac{18493488}{2977199}a^{12}-\frac{5502237}{2977199}a^{11}-\frac{64608192}{2977199}a^{10}+\frac{19378827}{2977199}a^{9}+\frac{152242828}{2977199}a^{8}-\frac{37448892}{2977199}a^{7}-\frac{234129696}{2977199}a^{6}+\frac{28228214}{2977199}a^{5}+\frac{204847344}{2977199}a^{4}+\frac{6254493}{2977199}a^{3}-\frac{58964544}{2977199}a^{2}-\frac{2330361}{2977199}a+\frac{667366}{2977199}$, $\frac{17}{2977199}a^{24}-\frac{1224}{2977199}a^{22}+\frac{38556}{2977199}a^{20}-\frac{222}{2977199}a^{19}-\frac{697680}{2977199}a^{18}+\frac{12654}{2977199}a^{17}+\frac{8005878}{2977199}a^{16}-\frac{304775}{2977199}a^{15}-\frac{60676846}{2977199}a^{14}+\frac{4034565}{2977199}a^{13}+\frac{306781692}{2977199}a^{12}-\frac{31964868}{2977199}a^{11}-\frac{1021298923}{2977199}a^{10}+\frac{154582857}{2977199}a^{9}+\frac{2157734595}{2977199}a^{8}-\frac{445219254}{2977199}a^{7}-\frac{2701301715}{2977199}a^{6}+\frac{708661583}{2977199}a^{5}+\frac{1776653082}{2977199}a^{4}-\frac{535985625}{2977199}a^{3}-\frac{489121659}{2977199}a^{2}+\frac{150664734}{2977199}a+\frac{22411478}{2977199}$, $\frac{3}{2977199}a^{26}-\frac{234}{2977199}a^{24}+\frac{8073}{2977199}a^{22}-\frac{162162}{2977199}a^{20}-\frac{31}{2977199}a^{19}+\frac{2100735}{2977199}a^{18}+\frac{1767}{2977199}a^{17}-\frac{18366307}{2977199}a^{16}-\frac{42408}{2977199}a^{15}+\frac{110192844}{2977199}a^{14}+\frac{562361}{2977199}a^{13}-\frac{452282688}{2977199}a^{12}-\frac{4566003}{2977199}a^{11}+\frac{1242937255}{2977199}a^{10}+\frac{23834421}{2977199}a^{9}-\frac{2188616070}{2977199}a^{8}-\frac{80922564}{2977199}a^{7}+\frac{2280909582}{2977199}a^{6}+\frac{169872390}{2977199}a^{5}-\frac{1207360620}{2977199}a^{4}-\frac{185336701}{2977199}a^{3}+\frac{221463720}{2977199}a^{2}+\frac{66933441}{2977199}a+\frac{92660}{50461}$, $\frac{599}{2977199}a^{16}+\frac{2876}{2977199}a^{15}-\frac{24079}{2977199}a^{14}-\frac{125465}{2977199}a^{13}+\frac{364398}{2977199}a^{12}+\frac{2175315}{2977199}a^{11}-\frac{2454507}{2977199}a^{10}-\frac{19040625}{2977199}a^{9}+\frac{5526630}{2977199}a^{8}+\frac{88171740}{2977199}a^{7}+\frac{13468518}{2977199}a^{6}-\frac{205867494}{2977199}a^{5}-\frac{75844188}{2977199}a^{4}+\frac{206067645}{2977199}a^{3}+\frac{83083401}{2977199}a^{2}-\frac{59842844}{2977199}a-\frac{18534022}{2977199}$, $\frac{61}{2977199}a^{22}-\frac{35}{2977199}a^{21}-\frac{4026}{2977199}a^{20}+\frac{2205}{2977199}a^{19}+\frac{114741}{2977199}a^{18}-\frac{59535}{2977199}a^{17}-\frac{1847934}{2977199}a^{16}+\frac{900358}{2977199}a^{15}+\frac{18482577}{2977199}a^{14}-\frac{8367210}{2977199}a^{13}-\frac{118838538}{2977199}a^{12}+\frac{49340745}{2977199}a^{11}+\frac{491891400}{2977199}a^{10}-\frac{184114755}{2977199}a^{9}-\frac{1277510182}{2977199}a^{8}+\frac{420374236}{2977199}a^{7}+\frac{1967195841}{2977199}a^{6}-\frac{546501333}{2977199}a^{5}-\frac{1617976746}{2977199}a^{4}+\frac{357341481}{2977199}a^{3}+\frac{577875978}{2977199}a^{2}-\frac{95205969}{2977199}a-\frac{51627382}{2977199}$, $\frac{5}{2977199}a^{25}-\frac{375}{2977199}a^{23}+\frac{48}{2977199}a^{22}+\frac{12375}{2977199}a^{21}-\frac{3168}{2977199}a^{20}-\frac{236250}{2977199}a^{19}+\frac{89942}{2977199}a^{18}+\frac{2885625}{2977199}a^{17}-\frac{1435428}{2977199}a^{16}-\frac{23546700}{2977199}a^{15}+\frac{14120730}{2977199}a^{14}+\frac{130126500}{2977199}a^{13}-\frac{88304580}{2977199}a^{12}-\frac{483352531}{2977199}a^{11}+\frac{349227450}{2977199}a^{10}+\frac{1173621273}{2977199}a^{9}-\frac{840936492}{2977199}a^{8}-\frac{1769111974}{2977199}a^{7}+\frac{1136548908}{2977199}a^{6}+\frac{1527285207}{2977199}a^{5}-\frac{734287559}{2977199}a^{4}-\frac{668498553}{2977199}a^{3}+\frac{155693994}{2977199}a^{2}+\frac{103219785}{2977199}a+\frac{7644940}{2977199}$, $\frac{4}{2977199}a^{27}+\frac{1}{2977199}a^{26}-\frac{340}{2977199}a^{25}-\frac{66}{2977199}a^{24}+\frac{12931}{2977199}a^{23}+\frac{1762}{2977199}a^{22}-\frac{290353}{2977199}a^{21}-\frac{22357}{2977199}a^{20}+\frac{4276467}{2977199}a^{19}+\frac{73374}{2977199}a^{18}-\frac{43402075}{2977199}a^{17}+\frac{1828048}{2977199}a^{16}+\frac{5257803}{50461}a^{15}-\frac{30803790}{2977199}a^{14}-\frac{1567226729}{2977199}a^{13}+\frac{238790471}{2977199}a^{12}+\frac{5530465607}{2977199}a^{11}-\frac{1098277697}{2977199}a^{10}-\frac{13234783173}{2977199}a^{9}+\frac{3096438581}{2977199}a^{8}+\frac{20335920464}{2977199}a^{7}-\frac{5116855280}{2977199}a^{6}-\frac{18182486726}{2977199}a^{5}+\frac{4308170166}{2977199}a^{4}+\frac{7811060934}{2977199}a^{3}-\frac{1185306545}{2977199}a^{2}-\frac{1047486680}{2977199}a-\frac{83407632}{2977199}$, $\frac{19}{2977199}a^{23}-\frac{1311}{2977199}a^{21}+\frac{39330}{2977199}a^{19}+\frac{253}{2977199}a^{18}-\frac{671944}{2977199}a^{17}-\frac{13662}{2977199}a^{16}+\frac{7190439}{2977199}a^{15}+\frac{307395}{2977199}a^{14}-\frac{49905387}{2977199}a^{13}-\frac{3717861}{2977199}a^{12}+\frac{224782742}{2977199}a^{11}+\frac{25947351}{2977199}a^{10}-\frac{638305932}{2977199}a^{9}-\frac{103789188}{2977199}a^{8}+\frac{1070680743}{2977199}a^{7}+\frac{220133629}{2977199}a^{6}-\frac{939150315}{2977199}a^{5}-\frac{204782841}{2977199}a^{4}+\frac{345289554}{2977199}a^{3}+\frac{61739820}{2977199}a^{2}-\frac{37503648}{2977199}a-\frac{7426208}{2977199}$, $\frac{1}{2977199}a^{26}-\frac{78}{2977199}a^{24}+\frac{13}{2977199}a^{23}+\frac{2691}{2977199}a^{22}-\frac{897}{2977199}a^{21}-\frac{53980}{2977199}a^{20}+\frac{26910}{2977199}a^{19}+\frac{695805}{2977199}a^{18}-\frac{460161}{2977199}a^{17}-\frac{6008922}{2977199}a^{16}+\frac{4940676}{2977199}a^{15}+\frac{35134452}{2977199}a^{14}-\frac{34584732}{2977199}a^{13}-\frac{137161674}{2977199}a^{12}+\frac{158682888}{2977199}a^{11}+\frac{342694638}{2977199}a^{10}-\frac{467533070}{2977199}a^{9}-\frac{500385600}{2977199}a^{8}+\frac{841188456}{2977199}a^{7}+\frac{341101464}{2977199}a^{6}-\frac{838007856}{2977199}a^{5}-\frac{17665398}{2977199}a^{4}+\frac{375475393}{2977199}a^{3}-\frac{59959764}{2977199}a^{2}-\frac{33037317}{2977199}a+\frac{2342270}{2977199}$, $\frac{6}{2977199}a^{27}+\frac{9}{2977199}a^{26}-\frac{476}{2977199}a^{25}-\frac{690}{2977199}a^{24}+\frac{16759}{2977199}a^{23}+\frac{23333}{2977199}a^{22}-\frac{345200}{2977199}a^{21}-\frac{457849}{2977199}a^{20}+\frac{4617244}{2977199}a^{19}+\frac{5769790}{2977199}a^{18}-\frac{42091111}{2977199}a^{17}-\frac{48805413}{2977199}a^{16}+\frac{267058497}{2977199}a^{15}+\frac{281209324}{2977199}a^{14}-\frac{1183025480}{2977199}a^{13}-\frac{1096396628}{2977199}a^{12}+\frac{3614528923}{2977199}a^{11}+\frac{2812354761}{2977199}a^{10}-\frac{7393910772}{2977199}a^{9}-\frac{4481365047}{2977199}a^{8}+\frac{9577654301}{2977199}a^{7}+\frac{3979408713}{2977199}a^{6}-\frac{7074278810}{2977199}a^{5}-\frac{1583385706}{2977199}a^{4}+\frac{2385090449}{2977199}a^{3}+\frac{192266599}{2977199}a^{2}-\frac{214777558}{2977199}a-\frac{30768454}{2977199}$, $\frac{1}{2977199}a^{27}+\frac{2}{2977199}a^{26}-\frac{86}{2977199}a^{25}-\frac{157}{2977199}a^{24}+\frac{3307}{2977199}a^{23}+\frac{5441}{2977199}a^{22}-\frac{74993}{2977199}a^{21}-\frac{109413}{2977199}a^{20}+\frac{1113885}{2977199}a^{19}+\frac{1410524}{2977199}a^{18}-\frac{11382824}{2977199}a^{17}-\frac{12146485}{2977199}a^{16}+\frac{81809725}{2977199}a^{15}+\frac{70491803}{2977199}a^{14}-\frac{415333103}{2977199}a^{13}-\frac{270675364}{2977199}a^{12}+\frac{1473246643}{2977199}a^{11}+\frac{650483098}{2977199}a^{10}-\frac{3547623662}{2977199}a^{9}-\frac{850699171}{2977199}a^{8}+\frac{5483619904}{2977199}a^{7}+\frac{341488598}{2977199}a^{6}-\frac{4884648530}{2977199}a^{5}+\frac{305960855}{2977199}a^{4}+\frac{1986299185}{2977199}a^{3}-\frac{155643943}{2977199}a^{2}-\frac{195070644}{2977199}a-\frac{16313811}{2977199}$ 2/2977199a^26 - 156/2977199a^24 + 5382/2977199a^22 - 108108/2977199a^20 + 1400490/2977199a^18 - 12244284/2977199a^16 + 73465704/2977199a^14 - 301596048/2977199a^12 + 829389132/2977199a^10 - 1463627880/2977199a^8 + 1536809274/2977199a^6 - 838259604/2977199a^4 + 933841/2977199a^3 + 179627058/2977199a^2 - 8404569/2977199a - 6377292/2977199, 16/2977199a^23 - 1104/2977199a^21 + 33120/2977199a^19 - 566352/2977199a^17 + 6080832/2977199a^15 - 42565824/2977199a^13 + 195302016/2977199a^11 - 575443440/2977199a^9 + 1035798192/2977199a^7 + 166427/2977199a^6 - 1035798192/2977199a^5 - 2995686/2977199a^4 + 478060704/2977199a^3 + 13480587/2977199a^2 - 65190096/2977199a - 8987058/2977199, 160/2977199a^17 - 8160/2977199a^15 + 171360/2977199a^13 - 1801/2977199a^12 - 1909440/2977199a^11 + 64836/2977199a^10 + 12117600/2977199a^9 - 875286/2977199a^8 - 43623360/2977199a^7 + 5446224/2977199a^6 + 83280960/2977199a^5 - 15317505/2977199a^4 - 71383680/2977199a^3 + 15755148/2977199a^2 + 17845920/2977199a - 2625858/2977199, 1/2977199a^24 - 72/2977199a^22 + 2268/2977199a^20 - 41040/2977199a^18 + 470934/2977199a^16 - 3569184/2977199a^14 + 18044208/2977199a^12 - 60046272/2977199a^10 + 126660105/2977199a^8 - 157621464/2977199a^6 + 217280/2977199a^5 + 101328084/2977199a^4 - 3259200/2977199a^3 - 25509168/2977199a^2 + 9777600/2977199a + 1062882/2977199, 16/2977199a^23 - 1104/2977199a^21 + 33120/2977199a^19 - 566352/2977199a^17 + 6080832/2977199a^15 - 42565824/2977199a^13 + 195302016/2977199a^11 - 575443440/2977199a^9 + 1035798192/2977199a^7 + 166427/2977199a^6 - 1035798192/2977199a^5 - 2995686/2977199a^4 + 478060704/2977199a^3 + 13480587/2977199a^2 - 65190096/2977199a - 6009859/2977199, 160/2977199a^17 - 8160/2977199a^15 + 171360/2977199a^13 - 1801/2977199a^12 - 1909440/2977199a^11 + 64836/2977199a^10 + 12117600/2977199a^9 - 875286/2977199a^8 - 43623360/2977199a^7 + 5446224/2977199a^6 + 83280960/2977199a^5 - 15317505/2977199a^4 - 71383680/2977199a^3 + 15755148/2977199a^2 + 17845920/2977199a + 351341/2977199, 1/2977199a^27 + 2/2977199a^26 - 86/2977199a^25 - 157/2977199a^24 + 3291/2977199a^23 + 5441/2977199a^22 - 73889/2977199a^21 - 109444/2977199a^20 + 1080481/2977199a^19 + 23943/50461a^18 - 10800284/2977199a^17 - 12207577/2977199a^16 + 75340381/2977199a^15 + 71468798/2977199a^14 - 367668059/2977199a^13 - 280117615/2977199a^12 + 1238180775/2977199a^11 + 706974062/2977199a^10 - 2785053131/2977199a^9 - 1055834689/2977199a^8 + 3933813595/2977199a^7 + 761553348/2977199a^6 - 3102179405/2977199a^5 - 95989324/2977199a^4 + 1059611071/2977199a^3 - 66112132/2977199a^2 - 84398481/2977199a - 4711513/2977199, 599/2977199a^16 - 28752/2977199a^14 + 3955/2977199a^13 + 560664/2977199a^12 - 154245/2977199a^11 - 5692896/2977199a^10 + 2313675/2977199a^9 + 32022540/2977199a^8 - 16658460/2977199a^7 - 97814304/2977199a^6 + 58304610/2977199a^5 + 146721456/2977199a^4 - 87456915/2977199a^3 - 83840832/2977199a^2 + 37481535/2977199a + 7860078/2977199, 1/2977199a^24 - 72/2977199a^22 + 35/2977199a^21 + 2268/2977199a^20 - 2205/2977199a^19 - 41040/2977199a^18 + 59535/2977199a^17 + 470934/2977199a^16 - 899640/2977199a^15 - 3569184/2977199a^14 + 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875286/2977199a^8 + 992174832/2977199a^7 + 5612651/2977199a^6 - 952517232/2977199a^5 - 18313191/2977199a^4 + 406677024/2977199a^3 + 29235735/2977199a^2 - 47344176/2977199a - 8635717/2977199, 1/2977199a^27 + 2/2977199a^26 - 81/2977199a^25 - 156/2977199a^24 + 2916/2977199a^23 + 5382/2977199a^22 - 61514/2977199a^21 - 108034/2977199a^20 + 844200/2977199a^19 + 1396050/2977199a^18 - 7913052/2977199a^17 - 12131064/2977199a^16 + 51759433/2977199a^15 + 71866586/2977199a^14 - 237160269/2977199a^13 - 287929542/2977199a^12 + 752575941/2977199a^11 + 756891630/2977199a^10 - 1606238495/2977199a^9 - 1228189606/2977199a^8 + 2178232047/2977199a^7 + 1093925526/2977199a^6 - 1680473061/2977199a^5 - 414516330/2977199a^4 + 579362455/2977199a^3 + 34643774/2977199a^2 - 48721185/2977199a - 3872892/2977199, 5/2977199a^25 - 375/2977199a^23 + 12375/2977199a^21 - 236250/2977199a^19 + 2885625/2977199a^17 - 23546700/2977199a^15 + 130126500/2977199a^13 - 483327000/2977199a^11 + 1172778750/2977199a^9 - 1759168125/2977199a^7 + 1477701225/2977199a^5 + 282001/2977199a^4 - 575727750/2977199a^3 - 3384012/2977199a^2 + 66430125/2977199a + 2098819/2977199, 1/2977199a^27 - 81/2977199a^25 + 2916/2977199a^23 - 61479/2977199a^21 - 43/2977199a^20 + 841995/2977199a^19 + 2420/2977199a^18 - 7853517/2977199a^17 - 57150/2977199a^16 + 50860872/2977199a^15 + 734400/2977199a^14 - 228873924/2977199a^13 - 5565105/2977199a^12 + 704696169/2977199a^11 + 25158276/2977199a^10 - 1435627813/2977199a^9 - 65194470/2977199a^8 + 1825189848/2977199a^7 + 86605200/2977199a^6 - 1309286106/2977199a^5 - 43794675/2977199a^4 + 433662984/2977199a^3 - 481498/2977199a^2 - 34397136/2977199a - 4207024/2977199, 4/2977199a^25 - 300/2977199a^23 + 9939/2977199a^21 - 191457/2977199a^19 + 2374999/2977199a^17 - 19847976/2977199a^15 + 113560020/2977199a^13 - 1801/2977199a^12 - 442902681/2977199a^11 + 64836/2977199a^10 + 1149556617/2977199a^9 - 892237/2977199a^8 - 1890046224/2977199a^7 + 5853048/2977199a^6 + 1797414381/2977199a^5 - 18738524/2977199a^4 - 827506125/2977199a^3 + 27516048/2977199a^2 + 119351151/2977199a - 17983420/2977199, 15/2977199a^24 - 1093/2977199a^22 + 34878/2977199a^20 - 640053/2977199a^18 - 919/2977199a^17 + 7457832/2977199a^16 + 46869/2977199a^15 - 57475980/2977199a^14 - 984249/2977199a^13 + 295952129/2977199a^12 + 10967346/2977199a^11 - 1004747859/2977199a^10 - 69600465/2977199a^9 + 2164217535/2977199a^8 + 250578625/2977199a^7 - 2741832315/2977199a^6 - 478418984/2977199a^5 + 1759258035/2977199a^4 + 407915823/2977199a^3 - 403236873/2977199a^2 - 93016197/2977199a + 2547200/2977199, 1/2977199a^24 - 72/2977199a^22 + 4/2977199a^21 + 2268/2977199a^20 - 252/2977199a^19 - 41040/2977199a^18 + 6804/2977199a^17 + 471414/2977199a^16 - 102816/2977199a^15 - 3592224/2977199a^14 + 950759/2977199a^13 + 18493488/2977199a^12 - 5502237/2977199a^11 - 64608192/2977199a^10 + 19378827/2977199a^9 + 152242828/2977199a^8 - 37448892/2977199a^7 - 234129696/2977199a^6 + 28228214/2977199a^5 + 204847344/2977199a^4 + 6254493/2977199a^3 - 58964544/2977199a^2 - 2330361/2977199a + 667366/2977199, 17/2977199a^24 - 1224/2977199a^22 + 38556/2977199a^20 - 222/2977199a^19 - 697680/2977199a^18 + 12654/2977199a^17 + 8005878/2977199a^16 - 304775/2977199a^15 - 60676846/2977199a^14 + 4034565/2977199a^13 + 306781692/2977199a^12 - 31964868/2977199a^11 - 1021298923/2977199a^10 + 154582857/2977199a^9 + 2157734595/2977199a^8 - 445219254/2977199a^7 - 2701301715/2977199a^6 + 708661583/2977199a^5 + 1776653082/2977199a^4 - 535985625/2977199a^3 - 489121659/2977199a^2 + 150664734/2977199a + 22411478/2977199, 3/2977199a^26 - 234/2977199a^24 + 8073/2977199a^22 - 162162/2977199a^20 - 31/2977199a^19 + 2100735/2977199a^18 + 1767/2977199a^17 - 18366307/2977199a^16 - 42408/2977199a^15 + 110192844/2977199a^14 + 562361/2977199a^13 - 452282688/2977199a^12 - 4566003/2977199a^11 + 1242937255/2977199a^10 + 23834421/2977199a^9 - 2188616070/2977199a^8 - 80922564/2977199a^7 + 2280909582/2977199a^6 + 169872390/2977199a^5 - 1207360620/2977199a^4 - 185336701/2977199a^3 + 221463720/2977199a^2 + 66933441/2977199a + 92660/50461, 599/2977199a^16 + 2876/2977199a^15 - 24079/2977199a^14 - 125465/2977199a^13 + 364398/2977199a^12 + 2175315/2977199a^11 - 2454507/2977199a^10 - 19040625/2977199a^9 + 5526630/2977199a^8 + 88171740/2977199a^7 + 13468518/2977199a^6 - 205867494/2977199a^5 - 75844188/2977199a^4 + 206067645/2977199a^3 + 83083401/2977199a^2 - 59842844/2977199a - 18534022/2977199, 61/2977199a^22 - 35/2977199a^21 - 4026/2977199a^20 + 2205/2977199a^19 + 114741/2977199a^18 - 59535/2977199a^17 - 1847934/2977199a^16 + 900358/2977199a^15 + 18482577/2977199a^14 - 8367210/2977199a^13 - 118838538/2977199a^12 + 49340745/2977199a^11 + 491891400/2977199a^10 - 184114755/2977199a^9 - 1277510182/2977199a^8 + 420374236/2977199a^7 + 1967195841/2977199a^6 - 546501333/2977199a^5 - 1617976746/2977199a^4 + 357341481/2977199a^3 + 577875978/2977199a^2 - 95205969/2977199a - 51627382/2977199, 5/2977199a^25 - 375/2977199a^23 + 48/2977199a^22 + 12375/2977199a^21 - 3168/2977199a^20 - 236250/2977199a^19 + 89942/2977199a^18 + 2885625/2977199a^17 - 1435428/2977199a^16 - 23546700/2977199a^15 + 14120730/2977199a^14 + 130126500/2977199a^13 - 88304580/2977199a^12 - 483352531/2977199a^11 + 349227450/2977199a^10 + 1173621273/2977199a^9 - 840936492/2977199a^8 - 1769111974/2977199a^7 + 1136548908/2977199a^6 + 1527285207/2977199a^5 - 734287559/2977199a^4 - 668498553/2977199a^3 + 155693994/2977199a^2 + 103219785/2977199a + 7644940/2977199, 4/2977199a^27 + 1/2977199a^26 - 340/2977199a^25 - 66/2977199a^24 + 12931/2977199a^23 + 1762/2977199a^22 - 290353/2977199a^21 - 22357/2977199a^20 + 4276467/2977199a^19 + 73374/2977199a^18 - 43402075/2977199a^17 + 1828048/2977199a^16 + 5257803/50461a^15 - 30803790/2977199a^14 - 1567226729/2977199a^13 + 238790471/2977199a^12 + 5530465607/2977199a^11 - 1098277697/2977199a^10 - 13234783173/2977199a^9 + 3096438581/2977199a^8 + 20335920464/2977199a^7 - 5116855280/2977199a^6 - 18182486726/2977199a^5 + 4308170166/2977199a^4 + 7811060934/2977199a^3 - 1185306545/2977199a^2 - 1047486680/2977199a - 83407632/2977199, 19/2977199a^23 - 1311/2977199a^21 + 39330/2977199a^19 + 253/2977199a^18 - 671944/2977199a^17 - 13662/2977199a^16 + 7190439/2977199a^15 + 307395/2977199a^14 - 49905387/2977199a^13 - 3717861/2977199a^12 + 224782742/2977199a^11 + 25947351/2977199a^10 - 638305932/2977199a^9 - 103789188/2977199a^8 + 1070680743/2977199a^7 + 220133629/2977199a^6 - 939150315/2977199a^5 - 204782841/2977199a^4 + 345289554/2977199a^3 + 61739820/2977199a^2 - 37503648/2977199a - 7426208/2977199, 1/2977199a^26 - 78/2977199a^24 + 13/2977199a^23 + 2691/2977199a^22 - 897/2977199a^21 - 53980/2977199a^20 + 26910/2977199a^19 + 695805/2977199a^18 - 460161/2977199a^17 - 6008922/2977199a^16 + 4940676/2977199a^15 + 35134452/2977199a^14 - 34584732/2977199a^13 - 137161674/2977199a^12 + 158682888/2977199a^11 + 342694638/2977199a^10 - 467533070/2977199a^9 - 500385600/2977199a^8 + 841188456/2977199a^7 + 341101464/2977199a^6 - 838007856/2977199a^5 - 17665398/2977199a^4 + 375475393/2977199a^3 - 59959764/2977199a^2 - 33037317/2977199a + 2342270/2977199, 6/2977199a^27 + 9/2977199a^26 - 476/2977199a^25 - 690/2977199a^24 + 16759/2977199a^23 + 23333/2977199a^22 - 345200/2977199a^21 - 457849/2977199a^20 + 4617244/2977199a^19 + 5769790/2977199a^18 - 42091111/2977199a^17 - 48805413/2977199a^16 + 267058497/2977199a^15 + 281209324/2977199a^14 - 1183025480/2977199a^13 - 1096396628/2977199a^12 + 3614528923/2977199a^11 + 2812354761/2977199a^10 - 7393910772/2977199a^9 - 4481365047/2977199a^8 + 9577654301/2977199a^7 + 3979408713/2977199a^6 - 7074278810/2977199a^5 - 1583385706/2977199a^4 + 2385090449/2977199a^3 + 192266599/2977199a^2 - 214777558/2977199a - 30768454/2977199, 1/2977199a^27 + 2/2977199a^26 - 86/2977199a^25 - 157/2977199a^24 + 3307/2977199a^23 + 5441/2977199a^22 - 74993/2977199a^21 - 109413/2977199a^20 + 1113885/2977199a^19 + 1410524/2977199a^18 - 11382824/2977199a^17 - 12146485/2977199a^16 + 81809725/2977199a^15 + 70491803/2977199a^14 - 415333103/2977199a^13 - 270675364/2977199a^12 + 1473246643/2977199a^11 + 650483098/2977199a^10 - 3547623662/2977199a^9 - 850699171/2977199a^8 + 5483619904/2977199a^7 + 341488598/2977199a^6 - 4884648530/2977199a^5 + 305960855/2977199a^4 + 1986299185/2977199a^3 - 155643943/2977199a^2 - 195070644/2977199a - 16313811/2977199 (assuming GRH)	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:	$ 1525104928454641200 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 1525104928454641200 \cdot 1}{2\cdot\sqrt{1159427335739550761098088865697701264851767903254485069}}\cr\approx \mathstrut & 0.190102505026846 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911)

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^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911, 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^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911);

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^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911);

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

$C_{28}$ (as 28T1):

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 28

The 28 conjugacy class representatives for $C_{28}$

Character table for $C_{28}$

Intermediate fields

$\Q(\sqrt{29}) $, 4.4.2951069.1, 7.7.594823321.1, $\Q(\zeta_{29})^+$

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	$28$	$28$	${\href{/padicField/5.14.0.1}{14} }^{2}$	${\href{/padicField/7.14.0.1}{14} }^{2}$	R	${\href{/padicField/13.7.0.1}{7} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{7}$	$28$	${\href{/padicField/23.7.0.1}{7} }^{4}$	R	$28$	$28$	${\href{/padicField/41.4.0.1}{4} }^{7}$	$28$	$28$	${\href{/padicField/53.7.0.1}{7} }^{4}$	${\href{/padicField/59.1.0.1}{1} }^{28}$

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
$11$ 11		Deg $28$	$2$	$14$	$14$
$29$ 29		Deg $28$	$28$	$1$	$27$

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