LMFDB - Number field 28.28.2070706293589565601613551437543564286910572644210741.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309)

gp: K = bnfinit(y^28 - y^27 - 57*y^26 + 57*y^25 + 1451*y^24 - 1451*y^23 - 21749*y^22 + 21749*y^21 + 213035*y^20 - 213035*y^19 - 1430453*y^18 + 1430453*y^17 + 6715531*y^16 - 6715531*y^15 - 22059893*y^14 + 22059893*y^13 + 49878667*y^12 - 49878667*y^11 - 74814837*y^10 + 74814837*y^9 + 69567115*y^8 - 69567115*y^7 - 35437941*y^6 + 35437941*y^5 + 7799435*y^4 - 7799435*y^3 - 515445*y^2 + 515445*y - 40309, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309)

$ x^{28} - x^{27} - 57 x^{26} + 57 x^{25} + 1451 x^{24} - 1451 x^{23} - 21749 x^{22} + 21749 x^{21} + \cdots - 40309 $ x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309

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:	$2070706293589565601613551437543564286910572644210741$ 2070706293589565601613551437543564286910572644210741 $\medspace = 7^{14}\cdot 29^{27}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$68.03$	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:	$7^{1/2}29^{27/28}\approx 68.03282703181608$
Ramified primes:	$7$, $29$ 7, 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:	$203=7\cdot 29$
Dirichlet character group:	$\lbrace$$\chi_{203}(64,·)$, $\chi_{203}(1,·)$, $\chi_{203}(195,·)$, $\chi_{203}(69,·)$, $\chi_{203}(71,·)$, $\chi_{203}(55,·)$, $\chi_{203}(76,·)$, $\chi_{203}(141,·)$, $\chi_{203}(78,·)$, $\chi_{203}(41,·)$, $\chi_{203}(22,·)$, $\chi_{203}(153,·)$, $\chi_{203}(90,·)$, $\chi_{203}(27,·)$, $\chi_{203}(92,·)$, $\chi_{203}(197,·)$, $\chi_{203}(160,·)$, $\chi_{203}(97,·)$, $\chi_{203}(36,·)$, $\chi_{203}(104,·)$, $\chi_{203}(169,·)$, $\chi_{203}(48,·)$, $\chi_{203}(118,·)$, $\chi_{203}(183,·)$, $\chi_{203}(120,·)$, $\chi_{203}(57,·)$, $\chi_{203}(188,·)$, $\chi_{203}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8641}a^{15}-\frac{4274}{8641}a^{14}-\frac{30}{8641}a^{13}-\frac{1302}{8641}a^{12}+\frac{360}{8641}a^{11}-\frac{2960}{8641}a^{10}-\frac{2200}{8641}a^{9}-\frac{351}{8641}a^{8}-\frac{1441}{8641}a^{7}+\frac{2711}{8641}a^{6}-\frac{3455}{8641}a^{5}+\frac{2146}{8641}a^{4}+\frac{319}{8641}a^{3}-\frac{1073}{8641}a^{2}-\frac{1920}{8641}a-\frac{3263}{8641}$, $\frac{1}{8641}a^{16}-\frac{32}{8641}a^{14}+\frac{93}{8641}a^{13}+\frac{416}{8641}a^{12}-\frac{2418}{8641}a^{11}-\frac{2816}{8641}a^{10}-\frac{1743}{8641}a^{9}+\frac{1919}{8641}a^{8}-\frac{3731}{8641}a^{7}-\frac{4222}{8641}a^{6}+\frac{2945}{8641}a^{5}+\frac{4222}{8641}a^{4}-\frac{2945}{8641}a^{3}+\frac{449}{8641}a^{2}-\frac{393}{8641}a+\frac{512}{8641}$, $\frac{1}{8641}a^{17}+\frac{1581}{8641}a^{14}-\frac{544}{8641}a^{13}-\frac{877}{8641}a^{12}+\frac{63}{8641}a^{11}-\frac{1412}{8641}a^{10}+\frac{647}{8641}a^{9}+\frac{2319}{8641}a^{8}+\frac{1512}{8641}a^{7}+\frac{3287}{8641}a^{6}-\frac{2646}{8641}a^{5}-\frac{3401}{8641}a^{4}+\frac{2016}{8641}a^{3}-\frac{165}{8641}a^{2}-\frac{441}{8641}a-\frac{724}{8641}$, $\frac{1}{8641}a^{18}-\frac{612}{8641}a^{14}+\frac{3348}{8641}a^{13}+\frac{1967}{8641}a^{12}-\frac{266}{8641}a^{11}-\frac{3015}{8641}a^{10}-\frac{1804}{8641}a^{9}+\frac{3419}{8641}a^{8}+\frac{284}{8641}a^{7}-\frac{2801}{8641}a^{6}-\frac{2158}{8641}a^{5}-\frac{3538}{8641}a^{4}-\frac{3326}{8641}a^{3}+\frac{2336}{8641}a^{2}+\frac{1805}{8641}a+\frac{126}{8641}$, $\frac{1}{8641}a^{19}-\frac{2758}{8641}a^{14}+\frac{889}{8641}a^{13}-\frac{2118}{8641}a^{12}+\frac{1280}{8641}a^{11}+\frac{1286}{8641}a^{10}-\frac{3626}{8641}a^{9}+\frac{1497}{8641}a^{8}-\frac{3311}{8641}a^{7}-\frac{2098}{8641}a^{6}-\frac{953}{8641}a^{5}-\frac{3406}{8641}a^{4}-\frac{1179}{8641}a^{3}+\frac{1845}{8641}a^{2}+\frac{262}{8641}a-\frac{885}{8641}$, $\frac{1}{8641}a^{20}-\frac{479}{8641}a^{14}+\frac{1552}{8641}a^{13}-\frac{3621}{8641}a^{12}+\frac{451}{8641}a^{11}-\frac{1561}{8641}a^{10}-\frac{121}{8641}a^{9}-\frac{3577}{8641}a^{8}-\frac{1516}{8641}a^{7}+\frac{1520}{8641}a^{6}-\frac{1273}{8641}a^{5}-\frac{1596}{8641}a^{4}+\frac{265}{8641}a^{3}-\frac{3850}{8641}a^{2}+\frac{688}{8641}a-\frac{4073}{8641}$, $\frac{1}{8641}a^{21}+\frac{2223}{8641}a^{14}-\frac{709}{8641}a^{13}-\frac{1055}{8641}a^{12}-\frac{1941}{8641}a^{11}-\frac{837}{8641}a^{10}-\frac{3175}{8641}a^{9}+\frac{3175}{8641}a^{8}+\frac{2561}{8641}a^{7}+\frac{1146}{8641}a^{6}+\frac{2531}{8641}a^{5}-\frac{80}{8641}a^{4}+\frac{2054}{8641}a^{3}-\frac{3460}{8641}a^{2}+\frac{834}{8641}a+\frac{1044}{8641}$, $\frac{1}{8641}a^{22}+\frac{3934}{8641}a^{14}-\frac{3493}{8641}a^{13}-\frac{2330}{8641}a^{12}+\frac{2496}{8641}a^{11}+\frac{1104}{8641}a^{10}+\frac{2969}{8641}a^{9}-\frac{3497}{8641}a^{8}-\frac{1322}{8641}a^{7}-\frac{1245}{8641}a^{6}-\frac{1464}{8641}a^{5}+\frac{1328}{8641}a^{4}-\frac{4035}{8641}a^{3}+\frac{1197}{8641}a^{2}+\frac{550}{8641}a+\frac{3850}{8641}$, $\frac{1}{8641}a^{23}+\frac{3678}{8641}a^{14}+\frac{3357}{8641}a^{13}+\frac{451}{8641}a^{12}+\frac{1988}{8641}a^{11}-\frac{459}{8641}a^{10}+\frac{1662}{8641}a^{9}-\frac{3048}{8641}a^{8}-\frac{847}{8641}a^{7}-\frac{3544}{8641}a^{6}+\frac{1005}{8641}a^{5}-\frac{4142}{8641}a^{4}-\frac{804}{8641}a^{3}-\frac{3717}{8641}a^{2}-\frac{3745}{8641}a-\frac{3884}{8641}$, $\frac{1}{8641}a^{24}-\frac{3491}{8641}a^{14}-\frac{1542}{8641}a^{13}+\frac{3630}{8641}a^{12}-\frac{2466}{8641}a^{11}+\frac{882}{8641}a^{10}+\frac{576}{8641}a^{9}+\frac{2622}{8641}a^{8}-\frac{479}{8641}a^{7}+\frac{1661}{8641}a^{6}+\frac{1078}{8641}a^{5}+\frac{4082}{8641}a^{4}-\frac{1823}{8641}a^{3}+\frac{2453}{8641}a^{2}-\frac{1821}{8641}a-\frac{1035}{8641}$, $\frac{1}{8641}a^{25}+\frac{931}{8641}a^{14}+\frac{2592}{8641}a^{13}-\frac{2582}{8641}a^{12}-\frac{3944}{8641}a^{11}+\frac{1852}{8641}a^{10}+\frac{4271}{8641}a^{9}+\frac{1202}{8641}a^{8}+\frac{192}{8641}a^{7}+\frac{3284}{8641}a^{6}-\frac{3128}{8641}a^{5}-\frac{1884}{8641}a^{4}+\frac{1393}{8641}a^{3}+\frac{2530}{8641}a^{2}+\frac{1661}{8641}a-\frac{2295}{8641}$, $\frac{1}{8641}a^{26}-\frac{1815}{8641}a^{14}-\frac{575}{8641}a^{13}-\frac{1522}{8641}a^{12}+\frac{3691}{8641}a^{11}+\frac{3552}{8641}a^{10}+\frac{1485}{8641}a^{9}-\frac{1385}{8641}a^{8}-\frac{3141}{8641}a^{7}-\frac{3897}{8641}a^{6}+\frac{269}{8641}a^{5}-\frac{462}{8641}a^{4}-\frac{665}{8641}a^{3}-\frac{1732}{8641}a^{2}-\frac{3462}{8641}a-\frac{3779}{8641}$, $\frac{1}{8641}a^{27}+\frac{1733}{8641}a^{14}-\frac{4126}{8641}a^{13}-\frac{446}{8641}a^{12}+\frac{236}{8641}a^{11}+\frac{3787}{8641}a^{10}-\frac{2243}{8641}a^{9}-\frac{772}{8641}a^{8}-\frac{1089}{8641}a^{7}+\frac{4005}{8641}a^{6}+\frac{2079}{8641}a^{5}-\frac{2766}{8641}a^{4}-\frac{1694}{8641}a^{3}+\frac{1909}{8641}a^{2}+\frac{2385}{8641}a-\frac{3260}{8641}$ 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, 1/8641*a^15 - 4274/8641*a^14 - 30/8641*a^13 - 1302/8641*a^12 + 360/8641*a^11 - 2960/8641*a^10 - 2200/8641*a^9 - 351/8641*a^8 - 1441/8641*a^7 + 2711/8641*a^6 - 3455/8641*a^5 + 2146/8641*a^4 + 319/8641*a^3 - 1073/8641*a^2 - 1920/8641*a - 3263/8641, 1/8641*a^16 - 32/8641*a^14 + 93/8641*a^13 + 416/8641*a^12 - 2418/8641*a^11 - 2816/8641*a^10 - 1743/8641*a^9 + 1919/8641*a^8 - 3731/8641*a^7 - 4222/8641*a^6 + 2945/8641*a^5 + 4222/8641*a^4 - 2945/8641*a^3 + 449/8641*a^2 - 393/8641*a + 512/8641, 1/8641*a^17 + 1581/8641*a^14 - 544/8641*a^13 - 877/8641*a^12 + 63/8641*a^11 - 1412/8641*a^10 + 647/8641*a^9 + 2319/8641*a^8 + 1512/8641*a^7 + 3287/8641*a^6 - 2646/8641*a^5 - 3401/8641*a^4 + 2016/8641*a^3 - 165/8641*a^2 - 441/8641*a - 724/8641, 1/8641*a^18 - 612/8641*a^14 + 3348/8641*a^13 + 1967/8641*a^12 - 266/8641*a^11 - 3015/8641*a^10 - 1804/8641*a^9 + 3419/8641*a^8 + 284/8641*a^7 - 2801/8641*a^6 - 2158/8641*a^5 - 3538/8641*a^4 - 3326/8641*a^3 + 2336/8641*a^2 + 1805/8641*a + 126/8641, 1/8641*a^19 - 2758/8641*a^14 + 889/8641*a^13 - 2118/8641*a^12 + 1280/8641*a^11 + 1286/8641*a^10 - 3626/8641*a^9 + 1497/8641*a^8 - 3311/8641*a^7 - 2098/8641*a^6 - 953/8641*a^5 - 3406/8641*a^4 - 1179/8641*a^3 + 1845/8641*a^2 + 262/8641*a - 885/8641, 1/8641*a^20 - 479/8641*a^14 + 1552/8641*a^13 - 3621/8641*a^12 + 451/8641*a^11 - 1561/8641*a^10 - 121/8641*a^9 - 3577/8641*a^8 - 1516/8641*a^7 + 1520/8641*a^6 - 1273/8641*a^5 - 1596/8641*a^4 + 265/8641*a^3 - 3850/8641*a^2 + 688/8641*a - 4073/8641, 1/8641*a^21 + 2223/8641*a^14 - 709/8641*a^13 - 1055/8641*a^12 - 1941/8641*a^11 - 837/8641*a^10 - 3175/8641*a^9 + 3175/8641*a^8 + 2561/8641*a^7 + 1146/8641*a^6 + 2531/8641*a^5 - 80/8641*a^4 + 2054/8641*a^3 - 3460/8641*a^2 + 834/8641*a + 1044/8641, 1/8641*a^22 + 3934/8641*a^14 - 3493/8641*a^13 - 2330/8641*a^12 + 2496/8641*a^11 + 1104/8641*a^10 + 2969/8641*a^9 - 3497/8641*a^8 - 1322/8641*a^7 - 1245/8641*a^6 - 1464/8641*a^5 + 1328/8641*a^4 - 4035/8641*a^3 + 1197/8641*a^2 + 550/8641*a + 3850/8641, 1/8641*a^23 + 3678/8641*a^14 + 3357/8641*a^13 + 451/8641*a^12 + 1988/8641*a^11 - 459/8641*a^10 + 1662/8641*a^9 - 3048/8641*a^8 - 847/8641*a^7 - 3544/8641*a^6 + 1005/8641*a^5 - 4142/8641*a^4 - 804/8641*a^3 - 3717/8641*a^2 - 3745/8641*a - 3884/8641, 1/8641*a^24 - 3491/8641*a^14 - 1542/8641*a^13 + 3630/8641*a^12 - 2466/8641*a^11 + 882/8641*a^10 + 576/8641*a^9 + 2622/8641*a^8 - 479/8641*a^7 + 1661/8641*a^6 + 1078/8641*a^5 + 4082/8641*a^4 - 1823/8641*a^3 + 2453/8641*a^2 - 1821/8641*a - 1035/8641, 1/8641*a^25 + 931/8641*a^14 + 2592/8641*a^13 - 2582/8641*a^12 - 3944/8641*a^11 + 1852/8641*a^10 + 4271/8641*a^9 + 1202/8641*a^8 + 192/8641*a^7 + 3284/8641*a^6 - 3128/8641*a^5 - 1884/8641*a^4 + 1393/8641*a^3 + 2530/8641*a^2 + 1661/8641*a - 2295/8641, 1/8641*a^26 - 1815/8641*a^14 - 575/8641*a^13 - 1522/8641*a^12 + 3691/8641*a^11 + 3552/8641*a^10 + 1485/8641*a^9 - 1385/8641*a^8 - 3141/8641*a^7 - 3897/8641*a^6 + 269/8641*a^5 - 462/8641*a^4 - 665/8641*a^3 - 1732/8641*a^2 - 3462/8641*a - 3779/8641, 1/8641*a^27 + 1733/8641*a^14 - 4126/8641*a^13 - 446/8641*a^12 + 236/8641*a^11 + 3787/8641*a^10 - 2243/8641*a^9 - 772/8641*a^8 - 1089/8641*a^7 + 4005/8641*a^6 + 2079/8641*a^5 - 2766/8641*a^4 - 1694/8641*a^3 + 1909/8641*a^2 + 2385/8641*a - 3260/8641

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:	$\frac{1}{8641}a^{16}-\frac{32}{8641}a^{14}+\frac{93}{8641}a^{13}+\frac{416}{8641}a^{12}-\frac{2418}{8641}a^{11}-\frac{2816}{8641}a^{10}+\frac{24180}{8641}a^{9}+\frac{10560}{8641}a^{8}-\frac{116064}{8641}a^{7}-\frac{21504}{8641}a^{6}+\frac{270816}{8641}a^{5}+\frac{21504}{8641}a^{4}-\frac{270816}{8641}a^{3}-\frac{8192}{8641}a^{2}+\frac{77376}{8641}a+\frac{512}{8641}$, $\frac{1}{8641}a^{26}-\frac{52}{8641}a^{24}+\frac{1196}{8641}a^{22}-\frac{16016}{8641}a^{20}+\frac{138320}{8641}a^{18}-\frac{806208}{8641}a^{16}+\frac{3224832}{8641}a^{14}-\frac{8825856}{8641}a^{12}+\frac{16180736}{8641}a^{10}-\frac{19036160}{8641}a^{8}+\frac{13325312}{8641}a^{6}-\frac{4845568}{8641}a^{4}+\frac{181}{8641}a^{3}+\frac{692224}{8641}a^{2}-\frac{1086}{8641}a-\frac{16384}{8641}$, $\frac{17}{8641}a^{20}-\frac{680}{8641}a^{18}+\frac{11560}{8641}a^{16}-\frac{108800}{8641}a^{14}+\frac{618800}{8641}a^{12}-\frac{2178176}{8641}a^{10}-\frac{627}{8641}a^{9}+\frac{4667520}{8641}a^{8}+\frac{11286}{8641}a^{7}-\frac{5744640}{8641}a^{6}-\frac{67716}{8641}a^{5}+\frac{3590400}{8641}a^{4}+\frac{150480}{8641}a^{3}-\frac{870400}{8641}a^{2}-\frac{90288}{8641}a+\frac{34816}{8641}$, $\frac{5}{8641}a^{23}-\frac{230}{8641}a^{21}+\frac{4600}{8641}a^{19}-\frac{52440}{8641}a^{17}+\frac{375360}{8641}a^{15}-\frac{1751680}{8641}a^{13}+\frac{5358080}{8641}a^{11}-\frac{10524800}{8641}a^{9}+\frac{12629760}{8641}a^{7}-\frac{967}{8641}a^{6}-\frac{8419840}{8641}a^{5}+\frac{11604}{8641}a^{4}+\frac{2590720}{8641}a^{3}-\frac{34812}{8641}a^{2}-\frac{235520}{8641}a+\frac{24113}{8641}$, $\frac{23}{8641}a^{18}-\frac{828}{8641}a^{16}+\frac{12420}{8641}a^{14}-\frac{100464}{8641}a^{12}-\frac{85}{8641}a^{11}+\frac{473616}{8641}a^{10}+\frac{1870}{8641}a^{9}-\frac{1311552}{8641}a^{8}-\frac{14960}{8641}a^{7}+\frac{2040192}{8641}a^{6}+\frac{52360}{8641}a^{5}-\frac{1589760}{8641}a^{4}-\frac{74800}{8641}a^{3}+\frac{476928}{8641}a^{2}+\frac{29920}{8641}a-\frac{23552}{8641}$, $\frac{3}{8641}a^{21}-\frac{126}{8641}a^{19}+\frac{2268}{8641}a^{17}-\frac{22848}{8641}a^{15}+\frac{141120}{8641}a^{13}-\frac{550368}{8641}a^{11}+\frac{1345344}{8641}a^{9}+\frac{287}{8641}a^{8}-\frac{1976832}{8641}a^{7}-\frac{4592}{8641}a^{6}+\frac{1596672}{8641}a^{5}+\frac{22960}{8641}a^{4}-\frac{591360}{8641}a^{3}-\frac{36736}{8641}a^{2}+\frac{64512}{8641}a+\frac{9184}{8641}$, $\frac{3}{8641}a^{25}-\frac{150}{8641}a^{23}+\frac{3300}{8641}a^{21}-\frac{42000}{8641}a^{19}+\frac{341955}{8641}a^{17}-\frac{1858950}{8641}a^{15}+\frac{6832980}{8641}a^{13}+\frac{271}{8641}a^{12}-\frac{16813680}{8641}a^{11}-\frac{6504}{8641}a^{10}+\frac{26782800}{8641}a^{9}+\frac{58536}{8641}a^{8}-\frac{25840320}{8641}a^{7}-\frac{242816}{8641}a^{6}+\frac{13319040}{8641}a^{5}+\frac{451231}{8641}a^{4}-\frac{2818560}{8641}a^{3}-\frac{279800}{8641}a^{2}+\frac{111360}{8641}a+\frac{2296}{8641}$, $\frac{1}{8641}a^{24}+\frac{5}{8641}a^{23}-\frac{48}{8641}a^{22}-\frac{230}{8641}a^{21}+\frac{1008}{8641}a^{20}+\frac{4600}{8641}a^{19}-\frac{12183}{8641}a^{18}-\frac{52395}{8641}a^{17}+\frac{93852}{8641}a^{16}+\frac{373830}{8641}a^{15}-\frac{482436}{8641}a^{14}-\frac{1730260}{8641}a^{13}+\frac{1684321}{8641}a^{12}+\frac{5199045}{8641}a^{11}-\frac{3981480}{8641}a^{10}-\frac{9853470}{8641}a^{9}+\frac{6195096}{8641}a^{8}+\frac{11029040}{8641}a^{7}-\frac{5898439}{8641}a^{6}-\frac{6417995}{8641}a^{5}+\frac{2903268}{8641}a^{4}+\frac{1511630}{8641}a^{3}-\frac{494460}{8641}a^{2}-\frac{111900}{8641}a+\frac{21169}{8641}$, $\frac{1}{8641}a^{27}+\frac{1}{8641}a^{26}-\frac{57}{8641}a^{25}-\frac{51}{8641}a^{24}+\frac{1451}{8641}a^{23}+\frac{1141}{8641}a^{22}-\frac{21749}{8641}a^{21}-\frac{14683}{8641}a^{20}+\frac{213035}{8641}a^{19}+\frac{119605}{8641}a^{18}-\frac{1430453}{8641}a^{17}-\frac{637963}{8641}a^{16}+\frac{6715531}{8641}a^{15}+\frac{2214773}{8641}a^{14}-\frac{22059893}{8641}a^{13}-\frac{4731019}{8641}a^{12}+\frac{49878667}{8641}a^{11}+\frac{5113205}{8641}a^{10}-\frac{74814837}{8641}a^{9}+\frac{134517}{8641}a^{8}+\frac{69567115}{8641}a^{7}-\frac{6308491}{8641}a^{6}-\frac{35437941}{8641}a^{5}+\frac{5246325}{8641}a^{4}+\frac{7799435}{8641}a^{3}-\frac{1090187}{8641}a^{2}-\frac{523362}{8641}a+\frac{64436}{8641}$, $\frac{1}{8641}a^{27}+\frac{1}{8641}a^{26}-\frac{54}{8641}a^{25}-\frac{52}{8641}a^{24}+\frac{1296}{8641}a^{23}+\frac{1196}{8641}a^{22}-\frac{18219}{8641}a^{21}-\frac{15999}{8641}a^{20}+\frac{166446}{8641}a^{19}+\frac{137640}{8641}a^{18}-\frac{1036476}{8641}a^{17}-\frac{794648}{8641}a^{16}+\frac{4487909}{8641}a^{15}+\frac{3116121}{8641}a^{14}-\frac{13533846}{8641}a^{13}-\frac{8209548}{8641}a^{12}+\frac{28013544}{8641}a^{11}+\frac{14029972}{8641}a^{10}-\frac{38485931}{8641}a^{9}-\frac{14518447}{8641}a^{8}+\frac{32962230}{8641}a^{7}+\frac{8003920}{8641}a^{6}-\frac{15659460}{8641}a^{5}-\frac{1836336}{8641}a^{4}+\frac{3281285}{8641}a^{3}+\frac{129385}{8641}a^{2}-\frac{202350}{8641}a+\frac{19580}{8641}$, $\frac{11}{8641}a^{19}-\frac{418}{8641}a^{17}+\frac{6688}{8641}a^{15}-\frac{58520}{8641}a^{13}+\frac{304304}{8641}a^{11}-\frac{457}{8641}a^{10}-\frac{956384}{8641}a^{9}+\frac{9140}{8641}a^{8}+\frac{1765632}{8641}a^{7}-\frac{63980}{8641}a^{6}-\frac{1765632}{8641}a^{5}+\frac{182800}{8641}a^{4}+\frac{802560}{8641}a^{3}-\frac{182800}{8641}a^{2}-\frac{107008}{8641}a+\frac{29248}{8641}$, $\frac{5}{8641}a^{23}-\frac{230}{8641}a^{21}+\frac{4600}{8641}a^{19}-\frac{52440}{8641}a^{17}+\frac{375360}{8641}a^{15}-\frac{1751680}{8641}a^{13}+\frac{5358080}{8641}a^{11}-\frac{10524800}{8641}a^{9}+\frac{12629760}{8641}a^{7}-\frac{967}{8641}a^{6}-\frac{8419840}{8641}a^{5}+\frac{11604}{8641}a^{4}+\frac{2590720}{8641}a^{3}-\frac{34812}{8641}a^{2}-\frac{235520}{8641}a+\frac{15472}{8641}$, $\frac{1}{8641}a^{27}+\frac{1}{8641}a^{26}-\frac{57}{8641}a^{25}-\frac{51}{8641}a^{24}+\frac{1451}{8641}a^{23}+\frac{1141}{8641}a^{22}-\frac{21749}{8641}a^{21}-\frac{14700}{8641}a^{20}+\frac{213046}{8641}a^{19}+\frac{120308}{8641}a^{18}-\frac{1430871}{8641}a^{17}-\frac{650351}{8641}a^{16}+\frac{6722219}{8641}a^{15}+\frac{2335993}{8641}a^{14}-\frac{22118413}{8641}a^{13}-\frac{5450283}{8641}a^{12}+\frac{50182886}{8641}a^{11}+\frac{7764540}{8641}a^{10}-\frac{75768724}{8641}a^{9}-\frac{5835415}{8641}a^{8}+\frac{71306501}{8641}a^{7}+\frac{1412361}{8641}a^{6}-\frac{37083497}{8641}a^{5}+\frac{248965}{8641}a^{4}+\frac{8376715}{8641}a^{3}+\frac{74341}{8641}a^{2}-\frac{510162}{8641}a+\frac{35316}{8641}$, $\frac{7}{8641}a^{22}-\frac{308}{8641}a^{20}+\frac{5897}{8641}a^{18}-\frac{64452}{8641}a^{16}+\frac{443180}{8641}a^{14}-\frac{1990352}{8641}a^{12}-\frac{542}{8641}a^{11}+\frac{5859568}{8641}a^{10}+\frac{11924}{8641}a^{9}-\frac{11022528}{8641}a^{8}-\frac{96933}{8641}a^{7}+\frac{12448128}{8641}a^{6}+\frac{355446}{8641}a^{5}-\frac{7447040}{8641}a^{4}-\frac{563256}{8641}a^{3}+\frac{1800448}{8641}a^{2}+\frac{277080}{8641}a-\frac{66111}{8641}$, $\frac{2}{8641}a^{23}-\frac{92}{8641}a^{21}+\frac{1840}{8641}a^{19}-\frac{21021}{8641}a^{17}+\frac{151674}{8641}a^{15}-\frac{722092}{8641}a^{13}+\frac{271}{8641}a^{12}+\frac{2302352}{8641}a^{11}-\frac{6504}{8641}a^{10}-\frac{4883120}{8641}a^{9}+\frac{58536}{8641}a^{8}+\frac{6667584}{8641}a^{7}-\frac{244931}{8641}a^{6}-\frac{5424256}{8641}a^{5}+\frac{480660}{8641}a^{4}+\frac{2211328}{8641}a^{3}-\frac{388332}{8641}a^{2}-\frac{290048}{8641}a+\frac{77169}{8641}$, $\frac{5}{8641}a^{23}-\frac{230}{8641}a^{21}+\frac{4600}{8641}a^{19}+\frac{23}{8641}a^{18}-\frac{52394}{8641}a^{17}-\frac{828}{8641}a^{16}+\frac{373796}{8641}a^{15}+\frac{12420}{8641}a^{14}-\frac{1729784}{8641}a^{13}-\frac{100549}{8641}a^{12}+\frac{5195339}{8641}a^{11}+\frac{475656}{8641}a^{10}-\frac{9834770}{8641}a^{9}-\frac{1329912}{8641}a^{8}+\frac{10963216}{8641}a^{7}+\frac{2115385}{8641}a^{6}-\frac{6265464}{8641}a^{5}-\frac{1720956}{8641}a^{4}+\frac{1314768}{8641}a^{3}+\frac{540036}{8641}a^{2}-\frac{5408}{8641}a-\frac{18960}{8641}$, $\frac{1}{8641}a^{27}-\frac{51}{8641}a^{25}+\frac{1146}{8641}a^{23}-\frac{14916}{8641}a^{21}+\frac{124320}{8641}a^{19}-\frac{692208}{8641}a^{17}+\frac{2604672}{8641}a^{15}-\frac{6541056}{8641}a^{13}+\frac{10523136}{8641}a^{11}-\frac{9884160}{8641}a^{9}+\frac{4173312}{8641}a^{7}+\frac{279552}{8641}a^{5}-\frac{4049}{8641}a^{4}-\frac{638976}{8641}a^{3}+\frac{32754}{8641}a^{2}+\frac{86016}{8641}a-\frac{25199}{8641}$, $\frac{3}{8641}a^{25}-\frac{150}{8641}a^{23}+\frac{3286}{8641}a^{21}-\frac{41412}{8641}a^{19}+\frac{331461}{8641}a^{17}-\frac{1755386}{8641}a^{15}+\frac{6217260}{8641}a^{13}-\frac{271}{8641}a^{12}-\frac{14563536}{8641}a^{11}+\frac{6504}{8641}a^{10}+\frac{21850928}{8641}a^{9}-\frac{56995}{8641}a^{8}-\frac{19846464}{8641}a^{7}+\frac{218160}{8641}a^{6}+\frac{9980544}{8641}a^{5}-\frac{336049}{8641}a^{4}-\frac{2408960}{8641}a^{3}+\frac{147336}{8641}a^{2}+\frac{201984}{8641}a-\frac{17768}{8641}$, $\frac{1}{8641}a^{26}-\frac{52}{8641}a^{24}+\frac{1196}{8641}a^{22}-\frac{16016}{8641}a^{20}+\frac{138320}{8641}a^{18}-\frac{806298}{8641}a^{16}+\frac{3227712}{8641}a^{14}+\frac{271}{8641}a^{13}-\frac{8863296}{8641}a^{12}-\frac{7046}{8641}a^{11}+\frac{16434176}{8641}a^{10}+\frac{70460}{8641}a^{9}-\frac{19986560}{8641}a^{8}-\frac{338208}{8641}a^{7}+\frac{15260672}{8641}a^{6}+\frac{789152}{8641}a^{5}-\frac{6780928}{8641}a^{4}-\frac{788971}{8641}a^{3}+\frac{1429504}{8641}a^{2}+\frac{224386}{8641}a-\frac{71105}{8641}$, $\frac{2}{8641}a^{27}+\frac{2}{8641}a^{26}-\frac{109}{8641}a^{25}-\frac{105}{8641}a^{24}+\frac{2645}{8641}a^{23}+\frac{2433}{8641}a^{22}-\frac{37673}{8641}a^{21}-\frac{32738}{8641}a^{20}+\frac{349503}{8641}a^{19}+\frac{283187}{8641}a^{18}-\frac{2215229}{8641}a^{17}-\frac{1646560}{8641}a^{16}+\frac{9783016}{8641}a^{15}+\frac{6535809}{8641}a^{14}-\frac{30124205}{8641}a^{13}-\frac{17620683}{8641}a^{12}+\frac{63621854}{8641}a^{11}+\frac{31467334}{8641}a^{10}-\frac{88838235}{8641}a^{9}-\frac{35395231}{8641}a^{8}+\frac{76719571}{8641}a^{7}+\frac{22964548}{8641}a^{6}-\frac{36310231}{8641}a^{5}-\frac{7470474}{8641}a^{4}+\frac{7405588}{8641}a^{3}+\frac{1044338}{8641}a^{2}-\frac{370761}{8641}a+\frac{18055}{8641}$, $\frac{1}{8641}a^{27}+\frac{2}{8641}a^{26}-\frac{53}{8641}a^{25}-\frac{103}{8641}a^{24}+\frac{1241}{8641}a^{23}+\frac{2334}{8641}a^{22}-\frac{16889}{8641}a^{21}-\frac{30584}{8641}a^{20}+\frac{147846}{8641}a^{19}+\frac{256120}{8641}a^{18}-\frac{870036}{8641}a^{17}-\frac{1429632}{8641}a^{16}+\frac{3492480}{8641}a^{15}+\frac{5381248}{8641}a^{14}-\frac{9500096}{8641}a^{13}-\frac{13505024}{8641}a^{12}+\frac{17030624}{8641}a^{11}+\frac{21800064}{8641}a^{10}-\frac{19008704}{8641}a^{9}-\frac{21049887}{8641}a^{8}+\frac{11825351}{8641}a^{7}+\frac{10475447}{8641}a^{6}-\frac{3163173}{8641}a^{5}-\frac{1777546}{8641}a^{4}+\frac{91087}{8641}a^{3}-\frac{52395}{8641}a^{2}+\frac{5446}{8641}a-\frac{355}{8641}$, $\frac{2}{8641}a^{26}-\frac{104}{8641}a^{24}-\frac{5}{8641}a^{23}+\frac{2392}{8641}a^{22}+\frac{230}{8641}a^{21}-\frac{32032}{8641}a^{20}-\frac{4600}{8641}a^{19}+\frac{276640}{8641}a^{18}+\frac{52440}{8641}a^{17}-\frac{1612416}{8641}a^{16}-\frac{375360}{8641}a^{15}+\frac{6449664}{8641}a^{14}+\frac{1751680}{8641}a^{13}-\frac{17651712}{8641}a^{12}-\frac{5358080}{8641}a^{11}+\frac{32361472}{8641}a^{10}+\frac{10524800}{8641}a^{9}-\frac{38072320}{8641}a^{8}-\frac{12629760}{8641}a^{7}+\frac{26651591}{8641}a^{6}+\frac{8419840}{8641}a^{5}-\frac{9702740}{8641}a^{4}-\frac{2598999}{8641}a^{3}+\frac{1419260}{8641}a^{2}+\frac{285194}{8641}a-\frac{39599}{8641}$, $\frac{6}{8641}a^{20}-\frac{263}{8641}a^{18}+\frac{4908}{8641}a^{16}-\frac{50820}{8641}a^{14}+\frac{318864}{8641}a^{12}+\frac{85}{8641}a^{11}-\frac{1242384}{8641}a^{10}-\frac{1583}{8641}a^{9}+\frac{2958912}{8641}a^{8}+\frac{9794}{8641}a^{7}-\frac{4067712}{8641}a^{6}-\frac{21364}{8641}a^{5}+\frac{2856960}{8641}a^{4}+\frac{5920}{8641}a^{3}-\frac{784128}{8641}a^{2}+\frac{11408}{8641}a+\frac{27199}{8641}$, $\frac{3}{8641}a^{26}-\frac{156}{8641}a^{24}+\frac{3588}{8641}a^{22}-\frac{48048}{8641}a^{20}+\frac{11}{8641}a^{19}+\frac{414960}{8641}a^{18}-\frac{418}{8641}a^{17}-\frac{2418623}{8641}a^{16}+\frac{6688}{8641}a^{15}+\frac{9674464}{8641}a^{14}-\frac{58427}{8641}a^{13}-\frac{26477152}{8641}a^{12}+\frac{301886}{8641}a^{11}+\frac{48538935}{8641}a^{10}-\frac{932204}{8641}a^{9}-\frac{57088780}{8641}a^{8}+\frac{1649568}{8641}a^{7}+\frac{39890452}{8641}a^{6}-\frac{1494816}{8641}a^{5}-\frac{14332400}{8641}a^{4}+\frac{523646}{8641}a^{3}+\frac{1885680}{8641}a^{2}+\frac{18956}{8641}a-\frac{19392}{8641}$, $\frac{1}{8641}a^{27}-\frac{2}{8641}a^{26}-\frac{53}{8641}a^{25}+\frac{109}{8641}a^{24}+\frac{1246}{8641}a^{23}-\frac{2642}{8641}a^{22}-\frac{17099}{8641}a^{21}+\frac{37535}{8641}a^{20}+\frac{151572}{8641}a^{19}-\frac{346720}{8641}a^{18}-\frac{905973}{8641}a^{17}+\frac{2182937}{8641}a^{16}+\frac{3691574}{8641}a^{15}-\frac{9545105}{8641}a^{14}-\frac{10081287}{8641}a^{13}+\frac{28964584}{8641}a^{12}+\frac{17390014}{8641}a^{11}-\frac{59837473}{8641}a^{10}-\frac{15827248}{8641}a^{9}+\frac{80642830}{8641}a^{8}+\frac{1145615}{8641}a^{7}-\frac{65323804}{8641}a^{6}+\frac{11088768}{8641}a^{5}+\frac{26898218}{8641}a^{4}-\frac{7913334}{8641}a^{3}-\frac{3578358}{8641}a^{2}+\frac{1256924}{8641}a-\frac{85466}{8641}$, $\frac{10}{8641}a^{22}-\frac{440}{8641}a^{20}+\frac{8360}{8641}a^{18}-\frac{89760}{8641}a^{16}+\frac{91}{8641}a^{15}+\frac{598311}{8641}a^{14}-\frac{2730}{8641}a^{13}-\frac{2560068}{8641}a^{12}+\frac{32760}{8641}a^{11}+\frac{7019628}{8641}a^{10}-\frac{200200}{8641}a^{9}-\frac{11931120}{8641}a^{8}+\frac{654233}{8641}a^{7}+\frac{11661984}{8641}a^{6}-\frac{1087198}{8641}a^{5}-\frac{5636992}{8641}a^{4}+\frac{761208}{8641}a^{3}+\frac{959936}{8641}a^{2}-\frac{120568}{8641}a-\frac{9535}{8641}$, $\frac{17}{8641}a^{21}-\frac{714}{8641}a^{19}+\frac{12852}{8641}a^{17}-\frac{1}{8641}a^{16}-\frac{129472}{8641}a^{15}+\frac{32}{8641}a^{14}+\frac{799587}{8641}a^{13}-\frac{416}{8641}a^{12}-\frac{3116334}{8641}a^{11}+\frac{2816}{8641}a^{10}+\frac{7599436}{8641}a^{9}-\frac{11814}{8641}a^{8}-\frac{11085984}{8641}a^{7}+\frac{41568}{8641}a^{6}+\frac{8776992}{8641}a^{5}-\frac{121824}{8641}a^{4}-\frac{3080224}{8641}a^{3}+\frac{168704}{8641}a^{2}+\frac{288192}{8641}a-\frac{31999}{8641}$ 1/8641a^16 - 32/8641a^14 + 93/8641a^13 + 416/8641a^12 - 2418/8641a^11 - 2816/8641a^10 + 24180/8641a^9 + 10560/8641a^8 - 116064/8641a^7 - 21504/8641a^6 + 270816/8641a^5 + 21504/8641a^4 - 270816/8641a^3 - 8192/8641a^2 + 77376/8641a + 512/8641, 1/8641a^26 - 52/8641a^24 + 1196/8641a^22 - 16016/8641a^20 + 138320/8641a^18 - 806208/8641a^16 + 3224832/8641a^14 - 8825856/8641a^12 + 16180736/8641a^10 - 19036160/8641a^8 + 13325312/8641a^6 - 4845568/8641a^4 + 181/8641a^3 + 692224/8641a^2 - 1086/8641a - 16384/8641, 17/8641a^20 - 680/8641a^18 + 11560/8641a^16 - 108800/8641a^14 + 618800/8641a^12 - 2178176/8641a^10 - 627/8641a^9 + 4667520/8641a^8 + 11286/8641a^7 - 5744640/8641a^6 - 67716/8641a^5 + 3590400/8641a^4 + 150480/8641a^3 - 870400/8641a^2 - 90288/8641a + 34816/8641, 5/8641a^23 - 230/8641a^21 + 4600/8641a^19 - 52440/8641a^17 + 375360/8641a^15 - 1751680/8641a^13 + 5358080/8641a^11 - 10524800/8641a^9 + 12629760/8641a^7 - 967/8641a^6 - 8419840/8641a^5 + 11604/8641a^4 + 2590720/8641a^3 - 34812/8641a^2 - 235520/8641a + 24113/8641, 23/8641a^18 - 828/8641a^16 + 12420/8641a^14 - 100464/8641a^12 - 85/8641a^11 + 473616/8641a^10 + 1870/8641a^9 - 1311552/8641a^8 - 14960/8641a^7 + 2040192/8641a^6 + 52360/8641a^5 - 1589760/8641a^4 - 74800/8641a^3 + 476928/8641a^2 + 29920/8641a - 23552/8641, 3/8641a^21 - 126/8641a^19 + 2268/8641a^17 - 22848/8641a^15 + 141120/8641a^13 - 550368/8641a^11 + 1345344/8641a^9 + 287/8641a^8 - 1976832/8641a^7 - 4592/8641a^6 + 1596672/8641a^5 + 22960/8641a^4 - 591360/8641a^3 - 36736/8641a^2 + 64512/8641a + 9184/8641, 3/8641a^25 - 150/8641a^23 + 3300/8641a^21 - 42000/8641a^19 + 341955/8641a^17 - 1858950/8641a^15 + 6832980/8641a^13 + 271/8641a^12 - 16813680/8641a^11 - 6504/8641a^10 + 26782800/8641a^9 + 58536/8641a^8 - 25840320/8641a^7 - 242816/8641a^6 + 13319040/8641a^5 + 451231/8641a^4 - 2818560/8641a^3 - 279800/8641a^2 + 111360/8641a + 2296/8641, 1/8641a^24 + 5/8641a^23 - 48/8641a^22 - 230/8641a^21 + 1008/8641a^20 + 4600/8641a^19 - 12183/8641a^18 - 52395/8641a^17 + 93852/8641a^16 + 373830/8641a^15 - 482436/8641a^14 - 1730260/8641a^13 + 1684321/8641a^12 + 5199045/8641a^11 - 3981480/8641a^10 - 9853470/8641a^9 + 6195096/8641a^8 + 11029040/8641a^7 - 5898439/8641a^6 - 6417995/8641a^5 + 2903268/8641a^4 + 1511630/8641a^3 - 494460/8641a^2 - 111900/8641a + 21169/8641, 1/8641a^27 + 1/8641a^26 - 57/8641a^25 - 51/8641a^24 + 1451/8641a^23 + 1141/8641a^22 - 21749/8641a^21 - 14683/8641a^20 + 213035/8641a^19 + 119605/8641a^18 - 1430453/8641a^17 - 637963/8641a^16 + 6715531/8641a^15 + 2214773/8641a^14 - 22059893/8641a^13 - 4731019/8641a^12 + 49878667/8641a^11 + 5113205/8641a^10 - 74814837/8641a^9 + 134517/8641a^8 + 69567115/8641a^7 - 6308491/8641a^6 - 35437941/8641a^5 + 5246325/8641a^4 + 7799435/8641a^3 - 1090187/8641a^2 - 523362/8641a + 64436/8641, 1/8641a^27 + 1/8641a^26 - 54/8641a^25 - 52/8641a^24 + 1296/8641a^23 + 1196/8641a^22 - 18219/8641a^21 - 15999/8641a^20 + 166446/8641a^19 + 137640/8641a^18 - 1036476/8641a^17 - 794648/8641a^16 + 4487909/8641a^15 + 3116121/8641a^14 - 13533846/8641a^13 - 8209548/8641a^12 + 28013544/8641a^11 + 14029972/8641a^10 - 38485931/8641a^9 - 14518447/8641a^8 + 32962230/8641a^7 + 8003920/8641a^6 - 15659460/8641a^5 - 1836336/8641a^4 + 3281285/8641a^3 + 129385/8641a^2 - 202350/8641a + 19580/8641, 11/8641a^19 - 418/8641a^17 + 6688/8641a^15 - 58520/8641a^13 + 304304/8641a^11 - 457/8641a^10 - 956384/8641a^9 + 9140/8641a^8 + 1765632/8641a^7 - 63980/8641a^6 - 1765632/8641a^5 + 182800/8641a^4 + 802560/8641a^3 - 182800/8641a^2 - 107008/8641a + 29248/8641, 5/8641a^23 - 230/8641a^21 + 4600/8641a^19 - 52440/8641a^17 + 375360/8641a^15 - 1751680/8641a^13 + 5358080/8641a^11 - 10524800/8641a^9 + 12629760/8641a^7 - 967/8641a^6 - 8419840/8641a^5 + 11604/8641a^4 + 2590720/8641a^3 - 34812/8641a^2 - 235520/8641a + 15472/8641, 1/8641a^27 + 1/8641a^26 - 57/8641a^25 - 51/8641a^24 + 1451/8641a^23 + 1141/8641a^22 - 21749/8641a^21 - 14700/8641a^20 + 213046/8641a^19 + 120308/8641a^18 - 1430871/8641a^17 - 650351/8641a^16 + 6722219/8641a^15 + 2335993/8641a^14 - 22118413/8641a^13 - 5450283/8641a^12 + 50182886/8641a^11 + 7764540/8641a^10 - 75768724/8641a^9 - 5835415/8641a^8 + 71306501/8641a^7 + 1412361/8641a^6 - 37083497/8641a^5 + 248965/8641a^4 + 8376715/8641a^3 + 74341/8641a^2 - 510162/8641a + 35316/8641, 7/8641a^22 - 308/8641a^20 + 5897/8641a^18 - 64452/8641a^16 + 443180/8641a^14 - 1990352/8641a^12 - 542/8641a^11 + 5859568/8641a^10 + 11924/8641a^9 - 11022528/8641a^8 - 96933/8641a^7 + 12448128/8641a^6 + 355446/8641a^5 - 7447040/8641a^4 - 563256/8641a^3 + 1800448/8641a^2 + 277080/8641a - 66111/8641, 2/8641a^23 - 92/8641a^21 + 1840/8641a^19 - 21021/8641a^17 + 151674/8641a^15 - 722092/8641a^13 + 271/8641a^12 + 2302352/8641a^11 - 6504/8641a^10 - 4883120/8641a^9 + 58536/8641a^8 + 6667584/8641a^7 - 244931/8641a^6 - 5424256/8641a^5 + 480660/8641a^4 + 2211328/8641a^3 - 388332/8641a^2 - 290048/8641a + 77169/8641, 5/8641a^23 - 230/8641a^21 + 4600/8641a^19 + 23/8641a^18 - 52394/8641a^17 - 828/8641a^16 + 373796/8641a^15 + 12420/8641a^14 - 1729784/8641a^13 - 100549/8641a^12 + 5195339/8641a^11 + 475656/8641a^10 - 9834770/8641a^9 - 1329912/8641a^8 + 10963216/8641a^7 + 2115385/8641a^6 - 6265464/8641a^5 - 1720956/8641a^4 + 1314768/8641a^3 + 540036/8641a^2 - 5408/8641a - 18960/8641, 1/8641a^27 - 51/8641a^25 + 1146/8641a^23 - 14916/8641a^21 + 124320/8641a^19 - 692208/8641a^17 + 2604672/8641a^15 - 6541056/8641a^13 + 10523136/8641a^11 - 9884160/8641a^9 + 4173312/8641a^7 + 279552/8641a^5 - 4049/8641a^4 - 638976/8641a^3 + 32754/8641a^2 + 86016/8641a - 25199/8641, 3/8641a^25 - 150/8641a^23 + 3286/8641a^21 - 41412/8641a^19 + 331461/8641a^17 - 1755386/8641a^15 + 6217260/8641a^13 - 271/8641a^12 - 14563536/8641a^11 + 6504/8641a^10 + 21850928/8641a^9 - 56995/8641a^8 - 19846464/8641a^7 + 218160/8641a^6 + 9980544/8641a^5 - 336049/8641a^4 - 2408960/8641a^3 + 147336/8641a^2 + 201984/8641a - 17768/8641, 1/8641a^26 - 52/8641a^24 + 1196/8641a^22 - 16016/8641a^20 + 138320/8641a^18 - 806298/8641a^16 + 3227712/8641a^14 + 271/8641a^13 - 8863296/8641a^12 - 7046/8641a^11 + 16434176/8641a^10 + 70460/8641a^9 - 19986560/8641a^8 - 338208/8641a^7 + 15260672/8641a^6 + 789152/8641a^5 - 6780928/8641a^4 - 788971/8641a^3 + 1429504/8641a^2 + 224386/8641a - 71105/8641, 2/8641a^27 + 2/8641a^26 - 109/8641a^25 - 105/8641a^24 + 2645/8641a^23 + 2433/8641a^22 - 37673/8641a^21 - 32738/8641a^20 + 349503/8641a^19 + 283187/8641a^18 - 2215229/8641a^17 - 1646560/8641a^16 + 9783016/8641a^15 + 6535809/8641a^14 - 30124205/8641a^13 - 17620683/8641a^12 + 63621854/8641a^11 + 31467334/8641a^10 - 88838235/8641a^9 - 35395231/8641a^8 + 76719571/8641a^7 + 22964548/8641a^6 - 36310231/8641a^5 - 7470474/8641a^4 + 7405588/8641a^3 + 1044338/8641a^2 - 370761/8641a + 18055/8641, 1/8641a^27 + 2/8641a^26 - 53/8641a^25 - 103/8641a^24 + 1241/8641a^23 + 2334/8641a^22 - 16889/8641a^21 - 30584/8641a^20 + 147846/8641a^19 + 256120/8641a^18 - 870036/8641a^17 - 1429632/8641a^16 + 3492480/8641a^15 + 5381248/8641a^14 - 9500096/8641a^13 - 13505024/8641a^12 + 17030624/8641a^11 + 21800064/8641a^10 - 19008704/8641a^9 - 21049887/8641a^8 + 11825351/8641a^7 + 10475447/8641a^6 - 3163173/8641a^5 - 1777546/8641a^4 + 91087/8641a^3 - 52395/8641a^2 + 5446/8641a - 355/8641, 2/8641a^26 - 104/8641a^24 - 5/8641a^23 + 2392/8641a^22 + 230/8641a^21 - 32032/8641a^20 - 4600/8641a^19 + 276640/8641a^18 + 52440/8641a^17 - 1612416/8641a^16 - 375360/8641a^15 + 6449664/8641a^14 + 1751680/8641a^13 - 17651712/8641a^12 - 5358080/8641a^11 + 32361472/8641a^10 + 10524800/8641a^9 - 38072320/8641a^8 - 12629760/8641a^7 + 26651591/8641a^6 + 8419840/8641a^5 - 9702740/8641a^4 - 2598999/8641a^3 + 1419260/8641a^2 + 285194/8641a - 39599/8641, 6/8641a^20 - 263/8641a^18 + 4908/8641a^16 - 50820/8641a^14 + 318864/8641a^12 + 85/8641a^11 - 1242384/8641a^10 - 1583/8641a^9 + 2958912/8641a^8 + 9794/8641a^7 - 4067712/8641a^6 - 21364/8641a^5 + 2856960/8641a^4 + 5920/8641a^3 - 784128/8641a^2 + 11408/8641a + 27199/8641, 3/8641a^26 - 156/8641a^24 + 3588/8641a^22 - 48048/8641a^20 + 11/8641a^19 + 414960/8641a^18 - 418/8641a^17 - 2418623/8641a^16 + 6688/8641a^15 + 9674464/8641a^14 - 58427/8641a^13 - 26477152/8641a^12 + 301886/8641a^11 + 48538935/8641a^10 - 932204/8641a^9 - 57088780/8641a^8 + 1649568/8641a^7 + 39890452/8641a^6 - 1494816/8641a^5 - 14332400/8641a^4 + 523646/8641a^3 + 1885680/8641a^2 + 18956/8641a - 19392/8641, 1/8641a^27 - 2/8641a^26 - 53/8641a^25 + 109/8641a^24 + 1246/8641a^23 - 2642/8641a^22 - 17099/8641a^21 + 37535/8641a^20 + 151572/8641a^19 - 346720/8641a^18 - 905973/8641a^17 + 2182937/8641a^16 + 3691574/8641a^15 - 9545105/8641a^14 - 10081287/8641a^13 + 28964584/8641a^12 + 17390014/8641a^11 - 59837473/8641a^10 - 15827248/8641a^9 + 80642830/8641a^8 + 1145615/8641a^7 - 65323804/8641a^6 + 11088768/8641a^5 + 26898218/8641a^4 - 7913334/8641a^3 - 3578358/8641a^2 + 1256924/8641a - 85466/8641, 10/8641a^22 - 440/8641a^20 + 8360/8641a^18 - 89760/8641a^16 + 91/8641a^15 + 598311/8641a^14 - 2730/8641a^13 - 2560068/8641a^12 + 32760/8641a^11 + 7019628/8641a^10 - 200200/8641a^9 - 11931120/8641a^8 + 654233/8641a^7 + 11661984/8641a^6 - 1087198/8641a^5 - 5636992/8641a^4 + 761208/8641a^3 + 959936/8641a^2 - 120568/8641a - 9535/8641, 17/8641a^21 - 714/8641a^19 + 12852/8641a^17 - 1/8641a^16 - 129472/8641a^15 + 32/8641a^14 + 799587/8641a^13 - 416/8641a^12 - 3116334/8641a^11 + 2816/8641a^10 + 7599436/8641a^9 - 11814/8641a^8 - 11085984/8641a^7 + 41568/8641a^6 + 8776992/8641a^5 - 121824/8641a^4 - 3080224/8641a^3 + 168704/8641a^2 + 288192/8641a - 31999/8641 (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:	$ 40658806140034130 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 40658806140034130 \cdot 1}{2\cdot\sqrt{2070706293589565601613551437543564286910572644210741}}\cr\approx \mathstrut & 0.119923764678369 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309)

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 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309, 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 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309);

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 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309);

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 $C_{28}$ is not computed

Intermediate fields

$\Q(\sqrt{29}) $, 4.4.1195061.1, 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	$28$	$28$	${\href{/padicField/5.7.0.1}{7} }^{4}$	R	$28$	${\href{/padicField/13.7.0.1}{7} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{7}$	$28$	${\href{/padicField/23.7.0.1}{7} }^{4}$	R	$28$	$28$	${\href{/padicField/41.4.0.1}{4} }^{7}$	$28$	$28$	${\href{/padicField/53.7.0.1}{7} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{14}$

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
$7$ 7	7.14.7.1	$x^{14} + 49 x^{12} + 1029 x^{10} + 12017 x^{8} + 8 x^{7} + 82859 x^{6} - 1176 x^{5} + 352947 x^{4} + 13720 x^{3} + 881203 x^{2} - 19160 x + 794999$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
$7$ 7	7.14.7.1	$x^{14} + 49 x^{12} + 1029 x^{10} + 12017 x^{8} + 8 x^{7} + 82859 x^{6} - 1176 x^{5} + 352947 x^{4} + 13720 x^{3} + 881203 x^{2} - 19160 x + 794999$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
$29$ 29		Deg $28$	$28$	$1$	$27$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more