sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1)
gp: K = bnfinit(y^27 - 11*y^26 + 70*y^25 - 273*y^24 + 723*y^23 - 1456*y^22 + 2649*y^21 - 4775*y^20 + 8022*y^19 - 11719*y^18 + 15552*y^17 - 20687*y^16 + 27099*y^15 - 31222*y^14 + 31020*y^13 - 30638*y^12 + 32802*y^11 - 31588*y^10 + 22446*y^9 - 12521*y^8 + 9384*y^7 - 8740*y^6 + 4644*y^5 - 254*y^4 - 460*y^3 - 287*y^2 + 245*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1)
x 27 − 11 x 26 + 70 x 25 − 273 x 24 + 723 x 23 − 1456 x 22 + 2649 x 21 − 4775 x 20 + ⋯ + 1 x^{27} - 11 x^{26} + 70 x^{25} - 273 x^{24} + 723 x^{23} - 1456 x^{22} + 2649 x^{21} - 4775 x^{20} + \cdots + 1 x 2 7 − 1 1 x 2 6 + 7 0 x 2 5 − 2 7 3 x 2 4 + 7 2 3 x 2 3 − 1 4 5 6 x 2 2 + 2 6 4 9 x 2 1 − 4 7 7 5 x 2 0 + ⋯ + 1
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : 27 27 2 7
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : [ 1 , 13 ] [1, 13] [ 1 , 1 3 ]
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
− 8138911451501750747538217172562287688025999 -8138911451501750747538217172562287688025999 − 8 1 3 8 9 1 1 4 5 1 5 0 1 7 5 0 7 4 7 5 3 8 2 1 7 1 7 2 5 6 2 2 8 7 6 8 8 0 2 5 9 9 9 = − 199 9 13 \medspace = -\,1999^{13} = − 1 9 9 9 1 3
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : 38.84 38.84 3 8 . 8 4
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 : 199 9 1 / 2 ≈ 44.710177812216315 1999^{1/2}\approx 44.710177812216315 1 9 9 9 1 / 2 ≈ 4 4 . 7 1 0 1 7 7 8 1 2 2 1 6 3 1 5 Ramified primes :
1999 1999 1 9 9 9
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : Q ( − 1999 ) \Q(\sqrt{-1999}) Q ( − 1 9 9 9 )
Aut ( K / Q ) \Aut(K/\Q) A u t ( K / Q ) C 1 C_1 C 1
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is not Galois over Q \Q Q This is not a CM field . This field has no CM subfields.
1 1 1 a a a a 2 a^{2} a 2 a 3 a^{3} a 3 a 4 a^{4} a 4 a 5 a^{5} a 5 a 6 a^{6} a 6 a 7 a^{7} a 7 1 3 a 8 − 1 3 \frac{1}{3}a^{8}-\frac{1}{3} 3 1 a 8 − 3 1 1 3 a 9 − 1 3 a \frac{1}{3}a^{9}-\frac{1}{3}a 3 1 a 9 − 3 1 a 1 3 a 10 − 1 3 a 2 \frac{1}{3}a^{10}-\frac{1}{3}a^{2} 3 1 a 1 0 − 3 1 a 2 1 3 a 11 − 1 3 a 3 \frac{1}{3}a^{11}-\frac{1}{3}a^{3} 3 1 a 1 1 − 3 1 a 3 1 3 a 12 − 1 3 a 4 \frac{1}{3}a^{12}-\frac{1}{3}a^{4} 3 1 a 1 2 − 3 1 a 4 1 3 a 13 − 1 3 a 5 \frac{1}{3}a^{13}-\frac{1}{3}a^{5} 3 1 a 1 3 − 3 1 a 5 1 3 a 14 − 1 3 a 6 \frac{1}{3}a^{14}-\frac{1}{3}a^{6} 3 1 a 1 4 − 3 1 a 6 1 3 a 15 − 1 3 a 7 \frac{1}{3}a^{15}-\frac{1}{3}a^{7} 3 1 a 1 5 − 3 1 a 7 1 9 a 16 + 1 9 a 8 − 2 9 \frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9} 9 1 a 1 6 + 9 1 a 8 − 9 2 1 9 a 17 + 1 9 a 9 − 2 9 a \frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a 9 1 a 1 7 + 9 1 a 9 − 9 2 a 1 9 a 18 + 1 9 a 10 − 2 9 a 2 \frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2} 9 1 a 1 8 + 9 1 a 1 0 − 9 2 a 2 1 9 a 19 + 1 9 a 11 − 2 9 a 3 \frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3} 9 1 a 1 9 + 9 1 a 1 1 − 9 2 a 3 1 27 a 20 + 1 27 a 19 + 1 27 a 18 − 1 27 a 17 + 1 27 a 16 + 1 9 a 15 + 1 9 a 14 − 2 27 a 12 + 4 27 a 11 − 2 27 a 10 + 2 27 a 9 + 4 27 a 8 − 1 9 a 7 − 1 9 a 6 + 1 27 a 4 − 5 27 a 3 + 1 27 a 2 − 1 27 a − 5 27 \frac{1}{27}a^{20}+\frac{1}{27}a^{19}+\frac{1}{27}a^{18}-\frac{1}{27}a^{17}+\frac{1}{27}a^{16}+\frac{1}{9}a^{15}+\frac{1}{9}a^{14}-\frac{2}{27}a^{12}+\frac{4}{27}a^{11}-\frac{2}{27}a^{10}+\frac{2}{27}a^{9}+\frac{4}{27}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{27}a^{4}-\frac{5}{27}a^{3}+\frac{1}{27}a^{2}-\frac{1}{27}a-\frac{5}{27} 2 7 1 a 2 0 + 2 7 1 a 1 9 + 2 7 1 a 1 8 − 2 7 1 a 1 7 + 2 7 1 a 1 6 + 9 1 a 1 5 + 9 1 a 1 4 − 2 7 2 a 1 2 + 2 7 4 a 1 1 − 2 7 2 a 1 0 + 2 7 2 a 9 + 2 7 4 a 8 − 9 1 a 7 − 9 1 a 6 + 2 7 1 a 4 − 2 7 5 a 3 + 2 7 1 a 2 − 2 7 1 a − 2 7 5 1 27 a 21 + 1 27 a 18 − 1 27 a 17 − 1 27 a 16 − 1 9 a 14 − 2 27 a 13 − 1 9 a 12 + 1 9 a 11 − 2 27 a 10 − 1 27 a 9 − 1 27 a 8 + 1 9 a 6 + 1 27 a 5 + 1 9 a 4 − 1 9 a 3 + 1 27 a 2 + 2 27 a + 2 27 \frac{1}{27}a^{21}+\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}-\frac{1}{9}a^{14}-\frac{2}{27}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{2}{27}a^{10}-\frac{1}{27}a^{9}-\frac{1}{27}a^{8}+\frac{1}{9}a^{6}+\frac{1}{27}a^{5}+\frac{1}{9}a^{4}-\frac{1}{9}a^{3}+\frac{1}{27}a^{2}+\frac{2}{27}a+\frac{2}{27} 2 7 1 a 2 1 + 2 7 1 a 1 8 − 2 7 1 a 1 7 − 2 7 1 a 1 6 − 9 1 a 1 4 − 2 7 2 a 1 3 − 9 1 a 1 2 + 9 1 a 1 1 − 2 7 2 a 1 0 − 2 7 1 a 9 − 2 7 1 a 8 + 9 1 a 6 + 2 7 1 a 5 + 9 1 a 4 − 9 1 a 3 + 2 7 1 a 2 + 2 7 2 a + 2 7 2 1 27 a 22 + 1 27 a 19 − 1 27 a 18 − 1 27 a 17 − 1 9 a 15 − 2 27 a 14 − 1 9 a 13 + 1 9 a 12 − 2 27 a 11 − 1 27 a 10 − 1 27 a 9 + 1 9 a 7 + 1 27 a 6 + 1 9 a 5 − 1 9 a 4 + 1 27 a 3 + 2 27 a 2 + 2 27 a \frac{1}{27}a^{22}+\frac{1}{27}a^{19}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{1}{9}a^{15}-\frac{2}{27}a^{14}-\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{2}{27}a^{11}-\frac{1}{27}a^{10}-\frac{1}{27}a^{9}+\frac{1}{9}a^{7}+\frac{1}{27}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{4}+\frac{1}{27}a^{3}+\frac{2}{27}a^{2}+\frac{2}{27}a 2 7 1 a 2 2 + 2 7 1 a 1 9 − 2 7 1 a 1 8 − 2 7 1 a 1 7 − 9 1 a 1 5 − 2 7 2 a 1 4 − 9 1 a 1 3 + 9 1 a 1 2 − 2 7 2 a 1 1 − 2 7 1 a 1 0 − 2 7 1 a 9 + 9 1 a 7 + 2 7 1 a 6 + 9 1 a 5 − 9 1 a 4 + 2 7 1 a 3 + 2 7 2 a 2 + 2 7 2 a 1 27 a 23 + 1 27 a 19 + 1 27 a 18 + 1 27 a 17 − 1 27 a 16 + 4 27 a 15 + 1 9 a 14 + 1 9 a 13 − 2 27 a 11 + 4 27 a 10 − 2 27 a 9 + 2 27 a 8 − 5 27 a 7 − 1 9 a 6 − 1 9 a 5 + 1 27 a 3 − 5 27 a 2 + 1 27 a − 1 27 \frac{1}{27}a^{23}+\frac{1}{27}a^{19}+\frac{1}{27}a^{18}+\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{4}{27}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{2}{27}a^{11}+\frac{4}{27}a^{10}-\frac{2}{27}a^{9}+\frac{2}{27}a^{8}-\frac{5}{27}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}+\frac{1}{27}a^{3}-\frac{5}{27}a^{2}+\frac{1}{27}a-\frac{1}{27} 2 7 1 a 2 3 + 2 7 1 a 1 9 + 2 7 1 a 1 8 + 2 7 1 a 1 7 − 2 7 1 a 1 6 + 2 7 4 a 1 5 + 9 1 a 1 4 + 9 1 a 1 3 − 2 7 2 a 1 1 + 2 7 4 a 1 0 − 2 7 2 a 9 + 2 7 2 a 8 − 2 7 5 a 7 − 9 1 a 6 − 9 1 a 5 + 2 7 1 a 3 − 2 7 5 a 2 + 2 7 1 a − 2 7 1 1 189 a 24 − 1 63 a 23 − 2 189 a 22 + 1 63 a 21 + 1 189 a 19 − 1 27 a 18 − 1 27 a 17 − 2 63 a 16 + 1 9 a 15 + 13 189 a 14 − 1 21 a 13 − 5 63 a 12 − 20 189 a 11 + 29 189 a 10 + 2 189 a 9 + 2 21 a 8 − 2 21 a 7 − 92 189 a 6 − 16 63 a 5 + 2 9 a 4 + 73 189 a 3 − 7 27 a 2 − 76 189 a + 41 189 \frac{1}{189}a^{24}-\frac{1}{63}a^{23}-\frac{2}{189}a^{22}+\frac{1}{63}a^{21}+\frac{1}{189}a^{19}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{2}{63}a^{16}+\frac{1}{9}a^{15}+\frac{13}{189}a^{14}-\frac{1}{21}a^{13}-\frac{5}{63}a^{12}-\frac{20}{189}a^{11}+\frac{29}{189}a^{10}+\frac{2}{189}a^{9}+\frac{2}{21}a^{8}-\frac{2}{21}a^{7}-\frac{92}{189}a^{6}-\frac{16}{63}a^{5}+\frac{2}{9}a^{4}+\frac{73}{189}a^{3}-\frac{7}{27}a^{2}-\frac{76}{189}a+\frac{41}{189} 1 8 9 1 a 2 4 − 6 3 1 a 2 3 − 1 8 9 2 a 2 2 + 6 3 1 a 2 1 + 1 8 9 1 a 1 9 − 2 7 1 a 1 8 − 2 7 1 a 1 7 − 6 3 2 a 1 6 + 9 1 a 1 5 + 1 8 9 1 3 a 1 4 − 2 1 1 a 1 3 − 6 3 5 a 1 2 − 1 8 9 2 0 a 1 1 + 1 8 9 2 9 a 1 0 + 1 8 9 2 a 9 + 2 1 2 a 8 − 2 1 2 a 7 − 1 8 9 9 2 a 6 − 6 3 1 6 a 5 + 9 2 a 4 + 1 8 9 7 3 a 3 − 2 7 7 a 2 − 1 8 9 7 6 a + 1 8 9 4 1 1 6968997 a 25 + 907 409941 a 24 − 119131 6968997 a 23 − 16438 2322999 a 22 + 111016 6968997 a 21 + 5534 774333 a 20 − 117550 6968997 a 19 + 5057 110619 a 18 + 281212 6968997 a 17 + 38770 774333 a 16 − 89272 6968997 a 15 − 34507 258111 a 14 + 994459 6968997 a 13 − 45130 331857 a 12 − 64678 6968997 a 11 − 113357 2322999 a 10 − 1043939 6968997 a 9 − 5744 110619 a 8 + 1188476 6968997 a 7 − 4358 331857 a 6 − 2482178 6968997 a 5 + 147064 2322999 a 4 − 60232 6968997 a 3 − 575689 2322999 a 2 + 43906 331857 a + 63727 6968997 \frac{1}{6968997}a^{25}+\frac{907}{409941}a^{24}-\frac{119131}{6968997}a^{23}-\frac{16438}{2322999}a^{22}+\frac{111016}{6968997}a^{21}+\frac{5534}{774333}a^{20}-\frac{117550}{6968997}a^{19}+\frac{5057}{110619}a^{18}+\frac{281212}{6968997}a^{17}+\frac{38770}{774333}a^{16}-\frac{89272}{6968997}a^{15}-\frac{34507}{258111}a^{14}+\frac{994459}{6968997}a^{13}-\frac{45130}{331857}a^{12}-\frac{64678}{6968997}a^{11}-\frac{113357}{2322999}a^{10}-\frac{1043939}{6968997}a^{9}-\frac{5744}{110619}a^{8}+\frac{1188476}{6968997}a^{7}-\frac{4358}{331857}a^{6}-\frac{2482178}{6968997}a^{5}+\frac{147064}{2322999}a^{4}-\frac{60232}{6968997}a^{3}-\frac{575689}{2322999}a^{2}+\frac{43906}{331857}a+\frac{63727}{6968997} 6 9 6 8 9 9 7 1 a 2 5 + 4 0 9 9 4 1 9 0 7 a 2 4 − 6 9 6 8 9 9 7 1 1 9 1 3 1 a 2 3 − 2 3 2 2 9 9 9 1 6 4 3 8 a 2 2 + 6 9 6 8 9 9 7 1 1 1 0 1 6 a 2 1 + 7 7 4 3 3 3 5 5 3 4 a 2 0 − 6 9 6 8 9 9 7 1 1 7 5 5 0 a 1 9 + 1 1 0 6 1 9 5 0 5 7 a 1 8 + 6 9 6 8 9 9 7 2 8 1 2 1 2 a 1 7 + 7 7 4 3 3 3 3 8 7 7 0 a 1 6 − 6 9 6 8 9 9 7 8 9 2 7 2 a 1 5 − 2 5 8 1 1 1 3 4 5 0 7 a 1 4 + 6 9 6 8 9 9 7 9 9 4 4 5 9 a 1 3 − 3 3 1 8 5 7 4 5 1 3 0 a 1 2 − 6 9 6 8 9 9 7 6 4 6 7 8 a 1 1 − 2 3 2 2 9 9 9 1 1 3 3 5 7 a 1 0 − 6 9 6 8 9 9 7 1 0 4 3 9 3 9 a 9 − 1 1 0 6 1 9 5 7 4 4 a 8 + 6 9 6 8 9 9 7 1 1 8 8 4 7 6 a 7 − 3 3 1 8 5 7 4 3 5 8 a 6 − 6 9 6 8 9 9 7 2 4 8 2 1 7 8 a 5 + 2 3 2 2 9 9 9 1 4 7 0 6 4 a 4 − 6 9 6 8 9 9 7 6 0 2 3 2 a 3 − 2 3 2 2 9 9 9 5 7 5 6 8 9 a 2 + 3 3 1 8 5 7 4 3 9 0 6 a + 6 9 6 8 9 9 7 6 3 7 2 7 1 79 ⋯ 03 a 26 + 10 ⋯ 46 26 ⋯ 01 a 25 + 12 ⋯ 97 79 ⋯ 03 a 24 − 68 ⋯ 04 46 ⋯ 59 a 23 − 39 ⋯ 03 79 ⋯ 03 a 22 − 13 ⋯ 39 79 ⋯ 03 a 21 + 50 ⋯ 68 79 ⋯ 03 a 20 + 36 ⋯ 43 79 ⋯ 03 a 19 − 28 ⋯ 31 79 ⋯ 03 a 18 + 30 ⋯ 54 79 ⋯ 03 a 17 − 22 ⋯ 59 79 ⋯ 03 a 16 − 86 ⋯ 95 79 ⋯ 03 a 15 − 26 ⋯ 31 79 ⋯ 03 a 14 − 40 ⋯ 81 79 ⋯ 03 a 13 − 78 ⋯ 42 79 ⋯ 03 a 12 + 43 ⋯ 51 79 ⋯ 03 a 11 − 28 ⋯ 08 79 ⋯ 03 a 10 + 20 ⋯ 53 16 ⋯ 47 a 9 − 99 ⋯ 34 79 ⋯ 03 a 8 − 47 ⋯ 42 11 ⋯ 29 a 7 − 30 ⋯ 29 42 ⋯ 37 a 6 + 22 ⋯ 85 42 ⋯ 37 a 5 + 10 ⋯ 88 79 ⋯ 03 a 4 + 13 ⋯ 33 79 ⋯ 03 a 3 − 19 ⋯ 94 88 ⋯ 67 a 2 − 14 ⋯ 44 42 ⋯ 37 a + 64 ⋯ 01 79 ⋯ 03 \frac{1}{79\cdots 03}a^{26}+\frac{10\cdots 46}{26\cdots 01}a^{25}+\frac{12\cdots 97}{79\cdots 03}a^{24}-\frac{68\cdots 04}{46\cdots 59}a^{23}-\frac{39\cdots 03}{79\cdots 03}a^{22}-\frac{13\cdots 39}{79\cdots 03}a^{21}+\frac{50\cdots 68}{79\cdots 03}a^{20}+\frac{36\cdots 43}{79\cdots 03}a^{19}-\frac{28\cdots 31}{79\cdots 03}a^{18}+\frac{30\cdots 54}{79\cdots 03}a^{17}-\frac{22\cdots 59}{79\cdots 03}a^{16}-\frac{86\cdots 95}{79\cdots 03}a^{15}-\frac{26\cdots 31}{79\cdots 03}a^{14}-\frac{40\cdots 81}{79\cdots 03}a^{13}-\frac{78\cdots 42}{79\cdots 03}a^{12}+\frac{43\cdots 51}{79\cdots 03}a^{11}-\frac{28\cdots 08}{79\cdots 03}a^{10}+\frac{20\cdots 53}{16\cdots 47}a^{9}-\frac{99\cdots 34}{79\cdots 03}a^{8}-\frac{47\cdots 42}{11\cdots 29}a^{7}-\frac{30\cdots 29}{42\cdots 37}a^{6}+\frac{22\cdots 85}{42\cdots 37}a^{5}+\frac{10\cdots 88}{79\cdots 03}a^{4}+\frac{13\cdots 33}{79\cdots 03}a^{3}-\frac{19\cdots 94}{88\cdots 67}a^{2}-\frac{14\cdots 44}{42\cdots 37}a+\frac{64\cdots 01}{79\cdots 03} 7 9 ⋯ 0 3 1 a 2 6 + 2 6 ⋯ 0 1 1 0 ⋯ 4 6 a 2 5 + 7 9 ⋯ 0 3 1 2 ⋯ 9 7 a 2 4 − 4 6 ⋯ 5 9 6 8 ⋯ 0 4 a 2 3 − 7 9 ⋯ 0 3 3 9 ⋯ 0 3 a 2 2 − 7 9 ⋯ 0 3 1 3 ⋯ 3 9 a 2 1 + 7 9 ⋯ 0 3 5 0 ⋯ 6 8 a 2 0 + 7 9 ⋯ 0 3 3 6 ⋯ 4 3 a 1 9 − 7 9 ⋯ 0 3 2 8 ⋯ 3 1 a 1 8 + 7 9 ⋯ 0 3 3 0 ⋯ 5 4 a 1 7 − 7 9 ⋯ 0 3 2 2 ⋯ 5 9 a 1 6 − 7 9 ⋯ 0 3 8 6 ⋯ 9 5 a 1 5 − 7 9 ⋯ 0 3 2 6 ⋯ 3 1 a 1 4 − 7 9 ⋯ 0 3 4 0 ⋯ 8 1 a 1 3 − 7 9 ⋯ 0 3 7 8 ⋯ 4 2 a 1 2 + 7 9 ⋯ 0 3 4 3 ⋯ 5 1 a 1 1 − 7 9 ⋯ 0 3 2 8 ⋯ 0 8 a 1 0 + 1 6 ⋯ 4 7 2 0 ⋯ 5 3 a 9 − 7 9 ⋯ 0 3 9 9 ⋯ 3 4 a 8 − 1 1 ⋯ 2 9 4 7 ⋯ 4 2 a 7 − 4 2 ⋯ 3 7 3 0 ⋯ 2 9 a 6 + 4 2 ⋯ 3 7 2 2 ⋯ 8 5 a 5 + 7 9 ⋯ 0 3 1 0 ⋯ 8 8 a 4 + 7 9 ⋯ 0 3 1 3 ⋯ 3 3 a 3 − 8 8 ⋯ 6 7 1 9 ⋯ 9 4 a 2 − 4 2 ⋯ 3 7 1 4 ⋯ 4 4 a + 7 9 ⋯ 0 3 6 4 ⋯ 0 1
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : C 5 C_{5} C 5 5 5 5 (assuming GRH )
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : C 5 C_{5} C 5 5 5 5 (assuming GRH )
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank : 13 13 1 3
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
− 1 -1 − 1 2 2 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 :
54 ⋯ 17 79 ⋯ 03 a 26 − 22 ⋯ 63 29 ⋯ 89 a 25 + 38 ⋯ 46 79 ⋯ 03 a 24 − 15 ⋯ 95 79 ⋯ 03 a 23 + 39 ⋯ 99 79 ⋯ 03 a 22 − 44 ⋯ 63 46 ⋯ 59 a 21 + 13 ⋯ 73 79 ⋯ 03 a 20 − 24 ⋯ 29 79 ⋯ 03 a 19 + 40 ⋯ 10 79 ⋯ 03 a 18 − 58 ⋯ 37 79 ⋯ 03 a 17 + 75 ⋯ 65 79 ⋯ 03 a 16 − 10 ⋯ 18 79 ⋯ 03 a 15 + 13 ⋯ 85 79 ⋯ 03 a 14 − 14 ⋯ 73 79 ⋯ 03 a 13 + 13 ⋯ 53 79 ⋯ 03 a 12 − 13 ⋯ 36 79 ⋯ 03 a 11 + 15 ⋯ 28 79 ⋯ 03 a 10 − 20 ⋯ 76 11 ⋯ 29 a 9 + 87 ⋯ 41 79 ⋯ 03 a 8 − 62 ⋯ 96 11 ⋯ 29 a 7 + 22 ⋯ 55 42 ⋯ 37 a 6 − 21 ⋯ 64 42 ⋯ 37 a 5 + 13 ⋯ 08 79 ⋯ 03 a 4 + 49 ⋯ 03 79 ⋯ 03 a 3 − 22 ⋯ 34 29 ⋯ 89 a 2 − 15 ⋯ 54 42 ⋯ 37 a + 77 ⋯ 56 79 ⋯ 03 \frac{54\cdots 17}{79\cdots 03}a^{26}-\frac{22\cdots 63}{29\cdots 89}a^{25}+\frac{38\cdots 46}{79\cdots 03}a^{24}-\frac{15\cdots 95}{79\cdots 03}a^{23}+\frac{39\cdots 99}{79\cdots 03}a^{22}-\frac{44\cdots 63}{46\cdots 59}a^{21}+\frac{13\cdots 73}{79\cdots 03}a^{20}-\frac{24\cdots 29}{79\cdots 03}a^{19}+\frac{40\cdots 10}{79\cdots 03}a^{18}-\frac{58\cdots 37}{79\cdots 03}a^{17}+\frac{75\cdots 65}{79\cdots 03}a^{16}-\frac{10\cdots 18}{79\cdots 03}a^{15}+\frac{13\cdots 85}{79\cdots 03}a^{14}-\frac{14\cdots 73}{79\cdots 03}a^{13}+\frac{13\cdots 53}{79\cdots 03}a^{12}-\frac{13\cdots 36}{79\cdots 03}a^{11}+\frac{15\cdots 28}{79\cdots 03}a^{10}-\frac{20\cdots 76}{11\cdots 29}a^{9}+\frac{87\cdots 41}{79\cdots 03}a^{8}-\frac{62\cdots 96}{11\cdots 29}a^{7}+\frac{22\cdots 55}{42\cdots 37}a^{6}-\frac{21\cdots 64}{42\cdots 37}a^{5}+\frac{13\cdots 08}{79\cdots 03}a^{4}+\frac{49\cdots 03}{79\cdots 03}a^{3}-\frac{22\cdots 34}{29\cdots 89}a^{2}-\frac{15\cdots 54}{42\cdots 37}a+\frac{77\cdots 56}{79\cdots 03} 7 9 ⋯ 0 3 5 4 ⋯ 1 7 a 2 6 − 2 9 ⋯ 8 9 2 2 ⋯ 6 3 a 2 5 + 7 9 ⋯ 0 3 3 8 ⋯ 4 6 a 2 4 − 7 9 ⋯ 0 3 1 5 ⋯ 9 5 a 2 3 + 7 9 ⋯ 0 3 3 9 ⋯ 9 9 a 2 2 − 4 6 ⋯ 5 9 4 4 ⋯ 6 3 a 2 1 + 7 9 ⋯ 0 3 1 3 ⋯ 7 3 a 2 0 − 7 9 ⋯ 0 3 2 4 ⋯ 2 9 a 1 9 + 7 9 ⋯ 0 3 4 0 ⋯ 1 0 a 1 8 − 7 9 ⋯ 0 3 5 8 ⋯ 3 7 a 1 7 + 7 9 ⋯ 0 3 7 5 ⋯ 6 5 a 1 6 − 7 9 ⋯ 0 3 1 0 ⋯ 1 8 a 1 5 + 7 9 ⋯ 0 3 1 3 ⋯ 8 5 a 1 4 − 7 9 ⋯ 0 3 1 4 ⋯ 7 3 a 1 3 + 7 9 ⋯ 0 3 1 3 ⋯ 5 3 a 1 2 − 7 9 ⋯ 0 3 1 3 ⋯ 3 6 a 1 1 + 7 9 ⋯ 0 3 1 5 ⋯ 2 8 a 1 0 − 1 1 ⋯ 2 9 2 0 ⋯ 7 6 a 9 + 7 9 ⋯ 0 3 8 7 ⋯ 4 1 a 8 − 1 1 ⋯ 2 9 6 2 ⋯ 9 6 a 7 + 4 2 ⋯ 3 7 2 2 ⋯ 5 5 a 6 − 4 2 ⋯ 3 7 2 1 ⋯ 6 4 a 5 + 7 9 ⋯ 0 3 1 3 ⋯ 0 8 a 4 + 7 9 ⋯ 0 3 4 9 ⋯ 0 3 a 3 − 2 9 ⋯ 8 9 2 2 ⋯ 3 4 a 2 − 4 2 ⋯ 3 7 1 5 ⋯ 5 4 a + 7 9 ⋯ 0 3 7 7 ⋯ 5 6 11 ⋯ 20 26 ⋯ 01 a 26 − 13 ⋯ 79 26 ⋯ 01 a 25 + 27 ⋯ 83 88 ⋯ 67 a 24 − 32 ⋯ 33 26 ⋯ 01 a 23 + 83 ⋯ 45 26 ⋯ 01 a 22 − 16 ⋯ 17 26 ⋯ 01 a 21 + 28 ⋯ 48 26 ⋯ 01 a 20 − 52 ⋯ 11 26 ⋯ 01 a 19 + 86 ⋯ 94 26 ⋯ 01 a 18 − 12 ⋯ 90 26 ⋯ 01 a 17 + 15 ⋯ 41 26 ⋯ 01 a 16 − 21 ⋯ 36 26 ⋯ 01 a 15 + 28 ⋯ 79 26 ⋯ 01 a 14 − 30 ⋯ 70 26 ⋯ 01 a 13 + 16 ⋯ 09 15 ⋯ 53 a 12 − 29 ⋯ 71 26 ⋯ 01 a 11 + 32 ⋯ 53 26 ⋯ 01 a 10 − 42 ⋯ 72 38 ⋯ 43 a 9 + 17 ⋯ 43 26 ⋯ 01 a 8 − 12 ⋯ 37 38 ⋯ 43 a 7 + 50 ⋯ 65 14 ⋯ 79 a 6 − 44 ⋯ 86 14 ⋯ 79 a 5 + 22 ⋯ 43 26 ⋯ 01 a 4 + 10 ⋯ 25 26 ⋯ 01 a 3 + 69 ⋯ 09 88 ⋯ 67 a 2 − 25 ⋯ 76 14 ⋯ 79 a + 24 ⋯ 09 26 ⋯ 01 \frac{11\cdots 20}{26\cdots 01}a^{26}-\frac{13\cdots 79}{26\cdots 01}a^{25}+\frac{27\cdots 83}{88\cdots 67}a^{24}-\frac{32\cdots 33}{26\cdots 01}a^{23}+\frac{83\cdots 45}{26\cdots 01}a^{22}-\frac{16\cdots 17}{26\cdots 01}a^{21}+\frac{28\cdots 48}{26\cdots 01}a^{20}-\frac{52\cdots 11}{26\cdots 01}a^{19}+\frac{86\cdots 94}{26\cdots 01}a^{18}-\frac{12\cdots 90}{26\cdots 01}a^{17}+\frac{15\cdots 41}{26\cdots 01}a^{16}-\frac{21\cdots 36}{26\cdots 01}a^{15}+\frac{28\cdots 79}{26\cdots 01}a^{14}-\frac{30\cdots 70}{26\cdots 01}a^{13}+\frac{16\cdots 09}{15\cdots 53}a^{12}-\frac{29\cdots 71}{26\cdots 01}a^{11}+\frac{32\cdots 53}{26\cdots 01}a^{10}-\frac{42\cdots 72}{38\cdots 43}a^{9}+\frac{17\cdots 43}{26\cdots 01}a^{8}-\frac{12\cdots 37}{38\cdots 43}a^{7}+\frac{50\cdots 65}{14\cdots 79}a^{6}-\frac{44\cdots 86}{14\cdots 79}a^{5}+\frac{22\cdots 43}{26\cdots 01}a^{4}+\frac{10\cdots 25}{26\cdots 01}a^{3}+\frac{69\cdots 09}{88\cdots 67}a^{2}-\frac{25\cdots 76}{14\cdots 79}a+\frac{24\cdots 09}{26\cdots 01} 2 6 ⋯ 0 1 1 1 ⋯ 2 0 a 2 6 − 2 6 ⋯ 0 1 1 3 ⋯ 7 9 a 2 5 + 8 8 ⋯ 6 7 2 7 ⋯ 8 3 a 2 4 − 2 6 ⋯ 0 1 3 2 ⋯ 3 3 a 2 3 + 2 6 ⋯ 0 1 8 3 ⋯ 4 5 a 2 2 − 2 6 ⋯ 0 1 1 6 ⋯ 1 7 a 2 1 + 2 6 ⋯ 0 1 2 8 ⋯ 4 8 a 2 0 − 2 6 ⋯ 0 1 5 2 ⋯ 1 1 a 1 9 + 2 6 ⋯ 0 1 8 6 ⋯ 9 4 a 1 8 − 2 6 ⋯ 0 1 1 2 ⋯ 9 0 a 1 7 + 2 6 ⋯ 0 1 1 5 ⋯ 4 1 a 1 6 − 2 6 ⋯ 0 1 2 1 ⋯ 3 6 a 1 5 + 2 6 ⋯ 0 1 2 8 ⋯ 7 9 a 1 4 − 2 6 ⋯ 0 1 3 0 ⋯ 7 0 a 1 3 + 1 5 ⋯ 5 3 1 6 ⋯ 0 9 a 1 2 − 2 6 ⋯ 0 1 2 9 ⋯ 7 1 a 1 1 + 2 6 ⋯ 0 1 3 2 ⋯ 5 3 a 1 0 − 3 8 ⋯ 4 3 4 2 ⋯ 7 2 a 9 + 2 6 ⋯ 0 1 1 7 ⋯ 4 3 a 8 − 3 8 ⋯ 4 3 1 2 ⋯ 3 7 a 7 + 1 4 ⋯ 7 9 5 0 ⋯ 6 5 a 6 − 1 4 ⋯ 7 9 4 4 ⋯ 8 6 a 5 + 2 6 ⋯ 0 1 2 2 ⋯ 4 3 a 4 + 2 6 ⋯ 0 1 1 0 ⋯ 2 5 a 3 + 8 8 ⋯ 6 7 6 9 ⋯ 0 9 a 2 − 1 4 ⋯ 7 9 2 5 ⋯ 7 6 a + 2 6 ⋯ 0 1 2 4 ⋯ 0 9 99 ⋯ 76 46 ⋯ 59 a 26 − 10 ⋯ 52 52 ⋯ 51 a 25 + 56 ⋯ 33 46 ⋯ 59 a 24 − 19 ⋯ 98 46 ⋯ 59 a 23 + 44 ⋯ 75 46 ⋯ 59 a 22 − 81 ⋯ 13 46 ⋯ 59 a 21 + 14 ⋯ 24 46 ⋯ 59 a 20 − 27 ⋯ 30 46 ⋯ 59 a 19 + 42 ⋯ 61 46 ⋯ 59 a 18 − 57 ⋯ 94 46 ⋯ 59 a 17 + 76 ⋯ 81 46 ⋯ 59 a 16 − 10 ⋯ 92 46 ⋯ 59 a 15 + 13 ⋯ 67 46 ⋯ 59 a 14 − 13 ⋯ 32 46 ⋯ 59 a 13 + 13 ⋯ 73 46 ⋯ 59 a 12 − 14 ⋯ 42 46 ⋯ 59 a 11 + 15 ⋯ 48 46 ⋯ 59 a 10 − 17 ⋯ 00 67 ⋯ 37 a 9 + 74 ⋯ 40 46 ⋯ 59 a 8 − 76 ⋯ 21 67 ⋯ 37 a 7 + 27 ⋯ 67 24 ⋯ 61 a 6 − 17 ⋯ 95 24 ⋯ 61 a 5 + 93 ⋯ 32 46 ⋯ 59 a 4 − 12 ⋯ 96 46 ⋯ 59 a 3 + 60 ⋯ 52 17 ⋯ 17 a 2 − 64 ⋯ 32 24 ⋯ 61 a + 56 ⋯ 44 46 ⋯ 59 \frac{99\cdots 76}{46\cdots 59}a^{26}-\frac{10\cdots 52}{52\cdots 51}a^{25}+\frac{56\cdots 33}{46\cdots 59}a^{24}-\frac{19\cdots 98}{46\cdots 59}a^{23}+\frac{44\cdots 75}{46\cdots 59}a^{22}-\frac{81\cdots 13}{46\cdots 59}a^{21}+\frac{14\cdots 24}{46\cdots 59}a^{20}-\frac{27\cdots 30}{46\cdots 59}a^{19}+\frac{42\cdots 61}{46\cdots 59}a^{18}-\frac{57\cdots 94}{46\cdots 59}a^{17}+\frac{76\cdots 81}{46\cdots 59}a^{16}-\frac{10\cdots 92}{46\cdots 59}a^{15}+\frac{13\cdots 67}{46\cdots 59}a^{14}-\frac{13\cdots 32}{46\cdots 59}a^{13}+\frac{13\cdots 73}{46\cdots 59}a^{12}-\frac{14\cdots 42}{46\cdots 59}a^{11}+\frac{15\cdots 48}{46\cdots 59}a^{10}-\frac{17\cdots 00}{67\cdots 37}a^{9}+\frac{74\cdots 40}{46\cdots 59}a^{8}-\frac{76\cdots 21}{67\cdots 37}a^{7}+\frac{27\cdots 67}{24\cdots 61}a^{6}-\frac{17\cdots 95}{24\cdots 61}a^{5}+\frac{93\cdots 32}{46\cdots 59}a^{4}-\frac{12\cdots 96}{46\cdots 59}a^{3}+\frac{60\cdots 52}{17\cdots 17}a^{2}-\frac{64\cdots 32}{24\cdots 61}a+\frac{56\cdots 44}{46\cdots 59} 4 6 ⋯ 5 9 9 9 ⋯ 7 6 a 2 6 − 5 2 ⋯ 5 1 1 0 ⋯ 5 2 a 2 5 + 4 6 ⋯ 5 9 5 6 ⋯ 3 3 a 2 4 − 4 6 ⋯ 5 9 1 9 ⋯ 9 8 a 2 3 + 4 6 ⋯ 5 9 4 4 ⋯ 7 5 a 2 2 − 4 6 ⋯ 5 9 8 1 ⋯ 1 3 a 2 1 + 4 6 ⋯ 5 9 1 4 ⋯ 2 4 a 2 0 − 4 6 ⋯ 5 9 2 7 ⋯ 3 0 a 1 9 + 4 6 ⋯ 5 9 4 2 ⋯ 6 1 a 1 8 − 4 6 ⋯ 5 9 5 7 ⋯ 9 4 a 1 7 + 4 6 ⋯ 5 9 7 6 ⋯ 8 1 a 1 6 − 4 6 ⋯ 5 9 1 0 ⋯ 9 2 a 1 5 + 4 6 ⋯ 5 9 1 3 ⋯ 6 7 a 1 4 − 4 6 ⋯ 5 9 1 3 ⋯ 3 2 a 1 3 + 4 6 ⋯ 5 9 1 3 ⋯ 7 3 a 1 2 − 4 6 ⋯ 5 9 1 4 ⋯ 4 2 a 1 1 + 4 6 ⋯ 5 9 1 5 ⋯ 4 8 a 1 0 − 6 7 ⋯ 3 7 1 7 ⋯ 0 0 a 9 + 4 6 ⋯ 5 9 7 4 ⋯ 4 0 a 8 − 6 7 ⋯ 3 7 7 6 ⋯ 2 1 a 7 + 2 4 ⋯ 6 1 2 7 ⋯ 6 7 a 6 − 2 4 ⋯ 6 1 1 7 ⋯ 9 5 a 5 + 4 6 ⋯ 5 9 9 3 ⋯ 3 2 a 4 − 4 6 ⋯ 5 9 1 2 ⋯ 9 6 a 3 + 1 7 ⋯ 1 7 6 0 ⋯ 5 2 a 2 − 2 4 ⋯ 6 1 6 4 ⋯ 3 2 a + 4 6 ⋯ 5 9 5 6 ⋯ 4 4 15 ⋯ 71 79 ⋯ 03 a 26 − 52 ⋯ 43 26 ⋯ 01 a 25 + 94 ⋯ 72 79 ⋯ 03 a 24 − 33 ⋯ 68 79 ⋯ 03 a 23 + 79 ⋯ 14 79 ⋯ 03 a 22 − 14 ⋯ 81 79 ⋯ 03 a 21 + 15 ⋯ 13 46 ⋯ 59 a 20 − 48 ⋯ 01 79 ⋯ 03 a 19 + 77 ⋯ 31 79 ⋯ 03 a 18 − 10 ⋯ 76 79 ⋯ 03 a 17 + 13 ⋯ 86 79 ⋯ 03 a 16 − 18 ⋯ 09 79 ⋯ 03 a 15 + 23 ⋯ 92 79 ⋯ 03 a 14 − 24 ⋯ 08 79 ⋯ 03 a 13 + 23 ⋯ 83 79 ⋯ 03 a 12 − 24 ⋯ 38 79 ⋯ 03 a 11 + 27 ⋯ 94 79 ⋯ 03 a 10 − 31 ⋯ 19 11 ⋯ 29 a 9 + 11 ⋯ 17 79 ⋯ 03 a 8 − 10 ⋯ 73 11 ⋯ 29 a 7 + 43 ⋯ 54 42 ⋯ 37 a 6 − 30 ⋯ 49 42 ⋯ 37 a 5 + 67 ⋯ 98 79 ⋯ 03 a 4 + 41 ⋯ 71 46 ⋯ 59 a 3 + 39 ⋯ 17 88 ⋯ 67 a 2 − 15 ⋯ 60 42 ⋯ 37 a − 60 ⋯ 01 79 ⋯ 03 \frac{15\cdots 71}{79\cdots 03}a^{26}-\frac{52\cdots 43}{26\cdots 01}a^{25}+\frac{94\cdots 72}{79\cdots 03}a^{24}-\frac{33\cdots 68}{79\cdots 03}a^{23}+\frac{79\cdots 14}{79\cdots 03}a^{22}-\frac{14\cdots 81}{79\cdots 03}a^{21}+\frac{15\cdots 13}{46\cdots 59}a^{20}-\frac{48\cdots 01}{79\cdots 03}a^{19}+\frac{77\cdots 31}{79\cdots 03}a^{18}-\frac{10\cdots 76}{79\cdots 03}a^{17}+\frac{13\cdots 86}{79\cdots 03}a^{16}-\frac{18\cdots 09}{79\cdots 03}a^{15}+\frac{23\cdots 92}{79\cdots 03}a^{14}-\frac{24\cdots 08}{79\cdots 03}a^{13}+\frac{23\cdots 83}{79\cdots 03}a^{12}-\frac{24\cdots 38}{79\cdots 03}a^{11}+\frac{27\cdots 94}{79\cdots 03}a^{10}-\frac{31\cdots 19}{11\cdots 29}a^{9}+\frac{11\cdots 17}{79\cdots 03}a^{8}-\frac{10\cdots 73}{11\cdots 29}a^{7}+\frac{43\cdots 54}{42\cdots 37}a^{6}-\frac{30\cdots 49}{42\cdots 37}a^{5}+\frac{67\cdots 98}{79\cdots 03}a^{4}+\frac{41\cdots 71}{46\cdots 59}a^{3}+\frac{39\cdots 17}{88\cdots 67}a^{2}-\frac{15\cdots 60}{42\cdots 37}a-\frac{60\cdots 01}{79\cdots 03} 7 9 ⋯ 0 3 1 5 ⋯ 7 1 a 2 6 − 2 6 ⋯ 0 1 5 2 ⋯ 4 3 a 2 5 + 7 9 ⋯ 0 3 9 4 ⋯ 7 2 a 2 4 − 7 9 ⋯ 0 3 3 3 ⋯ 6 8 a 2 3 + 7 9 ⋯ 0 3 7 9 ⋯ 1 4 a 2 2 − 7 9 ⋯ 0 3 1 4 ⋯ 8 1 a 2 1 + 4 6 ⋯ 5 9 1 5 ⋯ 1 3 a 2 0 − 7 9 ⋯ 0 3 4 8 ⋯ 0 1 a 1 9 + 7 9 ⋯ 0 3 7 7 ⋯ 3 1 a 1 8 − 7 9 ⋯ 0 3 1 0 ⋯ 7 6 a 1 7 + 7 9 ⋯ 0 3 1 3 ⋯ 8 6 a 1 6 − 7 9 ⋯ 0 3 1 8 ⋯ 0 9 a 1 5 + 7 9 ⋯ 0 3 2 3 ⋯ 9 2 a 1 4 − 7 9 ⋯ 0 3 2 4 ⋯ 0 8 a 1 3 + 7 9 ⋯ 0 3 2 3 ⋯ 8 3 a 1 2 − 7 9 ⋯ 0 3 2 4 ⋯ 3 8 a 1 1 + 7 9 ⋯ 0 3 2 7 ⋯ 9 4 a 1 0 − 1 1 ⋯ 2 9 3 1 ⋯ 1 9 a 9 + 7 9 ⋯ 0 3 1 1 ⋯ 1 7 a 8 − 1 1 ⋯ 2 9 1 0 ⋯ 7 3 a 7 + 4 2 ⋯ 3 7 4 3 ⋯ 5 4 a 6 − 4 2 ⋯ 3 7 3 0 ⋯ 4 9 a 5 + 7 9 ⋯ 0 3 6 7 ⋯ 9 8 a 4 + 4 6 ⋯ 5 9 4 1 ⋯ 7 1 a 3 + 8 8 ⋯ 6 7 3 9 ⋯ 1 7 a 2 − 4 2 ⋯ 3 7 1 5 ⋯ 6 0 a − 7 9 ⋯ 0 3 6 0 ⋯ 0 1 70 ⋯ 46 79 ⋯ 03 a 26 − 23 ⋯ 74 26 ⋯ 01 a 25 + 40 ⋯ 39 79 ⋯ 03 a 24 − 13 ⋯ 64 79 ⋯ 03 a 23 + 30 ⋯ 28 79 ⋯ 03 a 22 − 51 ⋯ 85 79 ⋯ 03 a 21 + 90 ⋯ 47 79 ⋯ 03 a 20 − 16 ⋯ 70 79 ⋯ 03 a 19 + 14 ⋯ 53 46 ⋯ 59 a 18 − 18 ⋯ 75 46 ⋯ 59 a 17 + 39 ⋯ 13 79 ⋯ 03 a 16 − 56 ⋯ 08 79 ⋯ 03 a 15 + 40 ⋯ 52 46 ⋯ 59 a 14 − 60 ⋯ 22 79 ⋯ 03 a 13 + 49 ⋯ 95 79 ⋯ 03 a 12 − 62 ⋯ 09 79 ⋯ 03 a 11 + 68 ⋯ 29 79 ⋯ 03 a 10 − 79 ⋯ 93 16 ⋯ 47 a 9 + 30 ⋯ 50 79 ⋯ 03 a 8 − 10 ⋯ 46 11 ⋯ 29 a 7 + 65 ⋯ 82 24 ⋯ 61 a 6 − 32 ⋯ 28 42 ⋯ 37 a 5 − 92 ⋯ 58 79 ⋯ 03 a 4 + 28 ⋯ 28 79 ⋯ 03 a 3 + 35 ⋯ 12 88 ⋯ 67 a 2 + 89 ⋯ 57 42 ⋯ 37 a − 62 ⋯ 87 79 ⋯ 03 \frac{70\cdots 46}{79\cdots 03}a^{26}-\frac{23\cdots 74}{26\cdots 01}a^{25}+\frac{40\cdots 39}{79\cdots 03}a^{24}-\frac{13\cdots 64}{79\cdots 03}a^{23}+\frac{30\cdots 28}{79\cdots 03}a^{22}-\frac{51\cdots 85}{79\cdots 03}a^{21}+\frac{90\cdots 47}{79\cdots 03}a^{20}-\frac{16\cdots 70}{79\cdots 03}a^{19}+\frac{14\cdots 53}{46\cdots 59}a^{18}-\frac{18\cdots 75}{46\cdots 59}a^{17}+\frac{39\cdots 13}{79\cdots 03}a^{16}-\frac{56\cdots 08}{79\cdots 03}a^{15}+\frac{40\cdots 52}{46\cdots 59}a^{14}-\frac{60\cdots 22}{79\cdots 03}a^{13}+\frac{49\cdots 95}{79\cdots 03}a^{12}-\frac{62\cdots 09}{79\cdots 03}a^{11}+\frac{68\cdots 29}{79\cdots 03}a^{10}-\frac{79\cdots 93}{16\cdots 47}a^{9}+\frac{30\cdots 50}{79\cdots 03}a^{8}-\frac{10\cdots 46}{11\cdots 29}a^{7}+\frac{65\cdots 82}{24\cdots 61}a^{6}-\frac{32\cdots 28}{42\cdots 37}a^{5}-\frac{92\cdots 58}{79\cdots 03}a^{4}+\frac{28\cdots 28}{79\cdots 03}a^{3}+\frac{35\cdots 12}{88\cdots 67}a^{2}+\frac{89\cdots 57}{42\cdots 37}a-\frac{62\cdots 87}{79\cdots 03} 7 9 ⋯ 0 3 7 0 ⋯ 4 6 a 2 6 − 2 6 ⋯ 0 1 2 3 ⋯ 7 4 a 2 5 + 7 9 ⋯ 0 3 4 0 ⋯ 3 9 a 2 4 − 7 9 ⋯ 0 3 1 3 ⋯ 6 4 a 2 3 + 7 9 ⋯ 0 3 3 0 ⋯ 2 8 a 2 2 − 7 9 ⋯ 0 3 5 1 ⋯ 8 5 a 2 1 + 7 9 ⋯ 0 3 9 0 ⋯ 4 7 a 2 0 − 7 9 ⋯ 0 3 1 6 ⋯ 7 0 a 1 9 + 4 6 ⋯ 5 9 1 4 ⋯ 5 3 a 1 8 − 4 6 ⋯ 5 9 1 8 ⋯ 7 5 a 1 7 + 7 9 ⋯ 0 3 3 9 ⋯ 1 3 a 1 6 − 7 9 ⋯ 0 3 5 6 ⋯ 0 8 a 1 5 + 4 6 ⋯ 5 9 4 0 ⋯ 5 2 a 1 4 − 7 9 ⋯ 0 3 6 0 ⋯ 2 2 a 1 3 + 7 9 ⋯ 0 3 4 9 ⋯ 9 5 a 1 2 − 7 9 ⋯ 0 3 6 2 ⋯ 0 9 a 1 1 + 7 9 ⋯ 0 3 6 8 ⋯ 2 9 a 1 0 − 1 6 ⋯ 4 7 7 9 ⋯ 9 3 a 9 + 7 9 ⋯ 0 3 3 0 ⋯ 5 0 a 8 − 1 1 ⋯ 2 9 1 0 ⋯ 4 6 a 7 + 2 4 ⋯ 6 1 6 5 ⋯ 8 2 a 6 − 4 2 ⋯ 3 7 3 2 ⋯ 2 8 a 5 − 7 9 ⋯ 0 3 9 2 ⋯ 5 8 a 4 + 7 9 ⋯ 0 3 2 8 ⋯ 2 8 a 3 + 8 8 ⋯ 6 7 3 5 ⋯ 1 2 a 2 + 4 2 ⋯ 3 7 8 9 ⋯ 5 7 a − 7 9 ⋯ 0 3 6 2 ⋯ 8 7 23 ⋯ 89 79 ⋯ 03 a 26 − 26 ⋯ 06 88 ⋯ 67 a 25 + 14 ⋯ 27 79 ⋯ 03 a 24 − 53 ⋯ 98 79 ⋯ 03 a 23 + 13 ⋯ 04 79 ⋯ 03 a 22 − 25 ⋯ 76 79 ⋯ 03 a 21 + 26 ⋯ 90 46 ⋯ 59 a 20 − 82 ⋯ 01 79 ⋯ 03 a 19 + 13 ⋯ 50 79 ⋯ 03 a 18 − 18 ⋯ 81 79 ⋯ 03 a 17 + 24 ⋯ 59 79 ⋯ 03 a 16 − 33 ⋯ 71 79 ⋯ 03 a 15 + 43 ⋯ 21 79 ⋯ 03 a 14 − 46 ⋯ 92 79 ⋯ 03 a 13 + 44 ⋯ 98 79 ⋯ 03 a 12 − 46 ⋯ 57 79 ⋯ 03 a 11 + 50 ⋯ 49 79 ⋯ 03 a 10 − 63 ⋯ 33 11 ⋯ 29 a 9 + 27 ⋯ 30 79 ⋯ 03 a 8 − 32 ⋯ 97 16 ⋯ 47 a 7 + 79 ⋯ 12 42 ⋯ 37 a 6 − 65 ⋯ 53 42 ⋯ 37 a 5 + 40 ⋯ 11 79 ⋯ 03 a 4 + 36 ⋯ 58 46 ⋯ 59 a 3 + 77 ⋯ 76 29 ⋯ 89 a 2 − 31 ⋯ 85 42 ⋯ 37 a + 11 ⋯ 46 79 ⋯ 03 \frac{23\cdots 89}{79\cdots 03}a^{26}-\frac{26\cdots 06}{88\cdots 67}a^{25}+\frac{14\cdots 27}{79\cdots 03}a^{24}-\frac{53\cdots 98}{79\cdots 03}a^{23}+\frac{13\cdots 04}{79\cdots 03}a^{22}-\frac{25\cdots 76}{79\cdots 03}a^{21}+\frac{26\cdots 90}{46\cdots 59}a^{20}-\frac{82\cdots 01}{79\cdots 03}a^{19}+\frac{13\cdots 50}{79\cdots 03}a^{18}-\frac{18\cdots 81}{79\cdots 03}a^{17}+\frac{24\cdots 59}{79\cdots 03}a^{16}-\frac{33\cdots 71}{79\cdots 03}a^{15}+\frac{43\cdots 21}{79\cdots 03}a^{14}-\frac{46\cdots 92}{79\cdots 03}a^{13}+\frac{44\cdots 98}{79\cdots 03}a^{12}-\frac{46\cdots 57}{79\cdots 03}a^{11}+\frac{50\cdots 49}{79\cdots 03}a^{10}-\frac{63\cdots 33}{11\cdots 29}a^{9}+\frac{27\cdots 30}{79\cdots 03}a^{8}-\frac{32\cdots 97}{16\cdots 47}a^{7}+\frac{79\cdots 12}{42\cdots 37}a^{6}-\frac{65\cdots 53}{42\cdots 37}a^{5}+\frac{40\cdots 11}{79\cdots 03}a^{4}+\frac{36\cdots 58}{46\cdots 59}a^{3}+\frac{77\cdots 76}{29\cdots 89}a^{2}-\frac{31\cdots 85}{42\cdots 37}a+\frac{11\cdots 46}{79\cdots 03} 7 9 ⋯ 0 3 2 3 ⋯ 8 9 a 2 6 − 8 8 ⋯ 6 7 2 6 ⋯ 0 6 a 2 5 + 7 9 ⋯ 0 3 1 4 ⋯ 2 7 a 2 4 − 7 9 ⋯ 0 3 5 3 ⋯ 9 8 a 2 3 + 7 9 ⋯ 0 3 1 3 ⋯ 0 4 a 2 2 − 7 9 ⋯ 0 3 2 5 ⋯ 7 6 a 2 1 + 4 6 ⋯ 5 9 2 6 ⋯ 9 0 a 2 0 − 7 9 ⋯ 0 3 8 2 ⋯ 0 1 a 1 9 + 7 9 ⋯ 0 3 1 3 ⋯ 5 0 a 1 8 − 7 9 ⋯ 0 3 1 8 ⋯ 8 1 a 1 7 + 7 9 ⋯ 0 3 2 4 ⋯ 5 9 a 1 6 − 7 9 ⋯ 0 3 3 3 ⋯ 7 1 a 1 5 + 7 9 ⋯ 0 3 4 3 ⋯ 2 1 a 1 4 − 7 9 ⋯ 0 3 4 6 ⋯ 9 2 a 1 3 + 7 9 ⋯ 0 3 4 4 ⋯ 9 8 a 1 2 − 7 9 ⋯ 0 3 4 6 ⋯ 5 7 a 1 1 + 7 9 ⋯ 0 3 5 0 ⋯ 4 9 a 1 0 − 1 1 ⋯ 2 9 6 3 ⋯ 3 3 a 9 + 7 9 ⋯ 0 3 2 7 ⋯ 3 0 a 8 − 1 6 ⋯ 4 7 3 2 ⋯ 9 7 a 7 + 4 2 ⋯ 3 7 7 9 ⋯ 1 2 a 6 − 4 2 ⋯ 3 7 6 5 ⋯ 5 3 a 5 + 7 9 ⋯ 0 3 4 0 ⋯ 1 1 a 4 + 4 6 ⋯ 5 9 3 6 ⋯ 5 8 a 3 + 2 9 ⋯ 8 9 7 7 ⋯ 7 6 a 2 − 4 2 ⋯ 3 7 3 1 ⋯ 8 5 a + 7 9 ⋯ 0 3 1 1 ⋯ 4 6 69 ⋯ 37 79 ⋯ 03 a 26 − 46 ⋯ 12 52 ⋯ 51 a 25 + 43 ⋯ 64 79 ⋯ 03 a 24 − 15 ⋯ 75 79 ⋯ 03 a 23 + 36 ⋯ 26 79 ⋯ 03 a 22 − 64 ⋯ 54 79 ⋯ 03 a 21 + 11 ⋯ 85 79 ⋯ 03 a 20 − 19 ⋯ 74 79 ⋯ 03 a 19 + 31 ⋯ 23 79 ⋯ 03 a 18 − 40 ⋯ 19 79 ⋯ 03 a 17 + 48 ⋯ 89 79 ⋯ 03 a 16 − 64 ⋯ 44 79 ⋯ 03 a 15 + 81 ⋯ 18 79 ⋯ 03 a 14 − 75 ⋯ 39 79 ⋯ 03 a 13 + 53 ⋯ 31 79 ⋯ 03 a 12 − 52 ⋯ 36 79 ⋯ 03 a 11 + 65 ⋯ 36 79 ⋯ 03 a 10 − 62 ⋯ 45 11 ⋯ 29 a 9 − 77 ⋯ 49 79 ⋯ 03 a 8 + 35 ⋯ 44 11 ⋯ 29 a 7 − 41 ⋯ 34 42 ⋯ 37 a 6 − 28 ⋯ 70 42 ⋯ 37 a 5 − 11 ⋯ 26 79 ⋯ 03 a 4 + 10 ⋯ 96 79 ⋯ 03 a 3 + 21 ⋯ 56 29 ⋯ 89 a 2 − 64 ⋯ 65 42 ⋯ 37 a − 62 ⋯ 11 46 ⋯ 59 \frac{69\cdots 37}{79\cdots 03}a^{26}-\frac{46\cdots 12}{52\cdots 51}a^{25}+\frac{43\cdots 64}{79\cdots 03}a^{24}-\frac{15\cdots 75}{79\cdots 03}a^{23}+\frac{36\cdots 26}{79\cdots 03}a^{22}-\frac{64\cdots 54}{79\cdots 03}a^{21}+\frac{11\cdots 85}{79\cdots 03}a^{20}-\frac{19\cdots 74}{79\cdots 03}a^{19}+\frac{31\cdots 23}{79\cdots 03}a^{18}-\frac{40\cdots 19}{79\cdots 03}a^{17}+\frac{48\cdots 89}{79\cdots 03}a^{16}-\frac{64\cdots 44}{79\cdots 03}a^{15}+\frac{81\cdots 18}{79\cdots 03}a^{14}-\frac{75\cdots 39}{79\cdots 03}a^{13}+\frac{53\cdots 31}{79\cdots 03}a^{12}-\frac{52\cdots 36}{79\cdots 03}a^{11}+\frac{65\cdots 36}{79\cdots 03}a^{10}-\frac{62\cdots 45}{11\cdots 29}a^{9}-\frac{77\cdots 49}{79\cdots 03}a^{8}+\frac{35\cdots 44}{11\cdots 29}a^{7}-\frac{41\cdots 34}{42\cdots 37}a^{6}-\frac{28\cdots 70}{42\cdots 37}a^{5}-\frac{11\cdots 26}{79\cdots 03}a^{4}+\frac{10\cdots 96}{79\cdots 03}a^{3}+\frac{21\cdots 56}{29\cdots 89}a^{2}-\frac{64\cdots 65}{42\cdots 37}a-\frac{62\cdots 11}{46\cdots 59} 7 9 ⋯ 0 3 6 9 ⋯ 3 7 a 2 6 − 5 2 ⋯ 5 1 4 6 ⋯ 1 2 a 2 5 + 7 9 ⋯ 0 3 4 3 ⋯ 6 4 a 2 4 − 7 9 ⋯ 0 3 1 5 ⋯ 7 5 a 2 3 + 7 9 ⋯ 0 3 3 6 ⋯ 2 6 a 2 2 − 7 9 ⋯ 0 3 6 4 ⋯ 5 4 a 2 1 + 7 9 ⋯ 0 3 1 1 ⋯ 8 5 a 2 0 − 7 9 ⋯ 0 3 1 9 ⋯ 7 4 a 1 9 + 7 9 ⋯ 0 3 3 1 ⋯ 2 3 a 1 8 − 7 9 ⋯ 0 3 4 0 ⋯ 1 9 a 1 7 + 7 9 ⋯ 0 3 4 8 ⋯ 8 9 a 1 6 − 7 9 ⋯ 0 3 6 4 ⋯ 4 4 a 1 5 + 7 9 ⋯ 0 3 8 1 ⋯ 1 8 a 1 4 − 7 9 ⋯ 0 3 7 5 ⋯ 3 9 a 1 3 + 7 9 ⋯ 0 3 5 3 ⋯ 3 1 a 1 2 − 7 9 ⋯ 0 3 5 2 ⋯ 3 6 a 1 1 + 7 9 ⋯ 0 3 6 5 ⋯ 3 6 a 1 0 − 1 1 ⋯ 2 9 6 2 ⋯ 4 5 a 9 − 7 9 ⋯ 0 3 7 7 ⋯ 4 9 a 8 + 1 1 ⋯ 2 9 3 5 ⋯ 4 4 a 7 − 4 2 ⋯ 3 7 4 1 ⋯ 3 4 a 6 − 4 2 ⋯ 3 7 2 8 ⋯ 7 0 a 5 − 7 9 ⋯ 0 3 1 1 ⋯ 2 6 a 4 + 7 9 ⋯ 0 3 1 0 ⋯ 9 6 a 3 + 2 9 ⋯ 8 9 2 1 ⋯ 5 6 a 2 − 4 2 ⋯ 3 7 6 4 ⋯ 6 5 a − 4 6 ⋯ 5 9 6 2 ⋯ 1 1 46 ⋯ 66 21 ⋯ 23 a 26 − 20 ⋯ 69 72 ⋯ 41 a 25 + 41 ⋯ 79 21 ⋯ 23 a 24 − 25 ⋯ 69 31 ⋯ 89 a 23 + 29 ⋯ 34 12 ⋯ 19 a 22 − 10 ⋯ 02 21 ⋯ 23 a 21 + 27 ⋯ 31 31 ⋯ 89 a 20 − 36 ⋯ 88 21 ⋯ 23 a 19 + 62 ⋯ 00 21 ⋯ 23 a 18 − 13 ⋯ 16 31 ⋯ 89 a 17 + 18 ⋯ 26 31 ⋯ 89 a 16 − 17 ⋯ 44 21 ⋯ 23 a 15 + 23 ⋯ 03 21 ⋯ 23 a 14 − 16 ⋯ 50 12 ⋯ 19 a 13 + 40 ⋯ 47 31 ⋯ 89 a 12 − 17 ⋯ 90 12 ⋯ 19 a 11 + 31 ⋯ 09 21 ⋯ 23 a 10 − 31 ⋯ 71 21 ⋯ 23 a 9 + 24 ⋯ 57 21 ⋯ 23 a 8 − 16 ⋯ 74 21 ⋯ 23 a 7 + 72 ⋯ 48 11 ⋯ 17 a 6 − 60 ⋯ 55 11 ⋯ 17 a 5 + 53 ⋯ 73 18 ⋯ 17 a 4 − 24 ⋯ 46 21 ⋯ 23 a 3 + 16 ⋯ 88 24 ⋯ 47 a 2 − 28 ⋯ 97 67 ⋯ 01 a + 27 ⋯ 02 21 ⋯ 23 \frac{46\cdots 66}{21\cdots 23}a^{26}-\frac{20\cdots 69}{72\cdots 41}a^{25}+\frac{41\cdots 79}{21\cdots 23}a^{24}-\frac{25\cdots 69}{31\cdots 89}a^{23}+\frac{29\cdots 34}{12\cdots 19}a^{22}-\frac{10\cdots 02}{21\cdots 23}a^{21}+\frac{27\cdots 31}{31\cdots 89}a^{20}-\frac{36\cdots 88}{21\cdots 23}a^{19}+\frac{62\cdots 00}{21\cdots 23}a^{18}-\frac{13\cdots 16}{31\cdots 89}a^{17}+\frac{18\cdots 26}{31\cdots 89}a^{16}-\frac{17\cdots 44}{21\cdots 23}a^{15}+\frac{23\cdots 03}{21\cdots 23}a^{14}-\frac{16\cdots 50}{12\cdots 19}a^{13}+\frac{40\cdots 47}{31\cdots 89}a^{12}-\frac{17\cdots 90}{12\cdots 19}a^{11}+\frac{31\cdots 09}{21\cdots 23}a^{10}-\frac{31\cdots 71}{21\cdots 23}a^{9}+\frac{24\cdots 57}{21\cdots 23}a^{8}-\frac{16\cdots 74}{21\cdots 23}a^{7}+\frac{72\cdots 48}{11\cdots 17}a^{6}-\frac{60\cdots 55}{11\cdots 17}a^{5}+\frac{53\cdots 73}{18\cdots 17}a^{4}-\frac{24\cdots 46}{21\cdots 23}a^{3}+\frac{16\cdots 88}{24\cdots 47}a^{2}-\frac{28\cdots 97}{67\cdots 01}a+\frac{27\cdots 02}{21\cdots 23} 2 1 ⋯ 2 3 4 6 ⋯ 6 6 a 2 6 − 7 2 ⋯ 4 1 2 0 ⋯ 6 9 a 2 5 + 2 1 ⋯ 2 3 4 1 ⋯ 7 9 a 2 4 − 3 1 ⋯ 8 9 2 5 ⋯ 6 9 a 2 3 + 1 2 ⋯ 1 9 2 9 ⋯ 3 4 a 2 2 − 2 1 ⋯ 2 3 1 0 ⋯ 0 2 a 2 1 + 3 1 ⋯ 8 9 2 7 ⋯ 3 1 a 2 0 − 2 1 ⋯ 2 3 3 6 ⋯ 8 8 a 1 9 + 2 1 ⋯ 2 3 6 2 ⋯ 0 0 a 1 8 − 3 1 ⋯ 8 9 1 3 ⋯ 1 6 a 1 7 + 3 1 ⋯ 8 9 1 8 ⋯ 2 6 a 1 6 − 2 1 ⋯ 2 3 1 7 ⋯ 4 4 a 1 5 + 2 1 ⋯ 2 3 2 3 ⋯ 0 3 a 1 4 − 1 2 ⋯ 1 9 1 6 ⋯ 5 0 a 1 3 + 3 1 ⋯ 8 9 4 0 ⋯ 4 7 a 1 2 − 1 2 ⋯ 1 9 1 7 ⋯ 9 0 a 1 1 + 2 1 ⋯ 2 3 3 1 ⋯ 0 9 a 1 0 − 2 1 ⋯ 2 3 3 1 ⋯ 7 1 a 9 + 2 1 ⋯ 2 3 2 4 ⋯ 5 7 a 8 − 2 1 ⋯ 2 3 1 6 ⋯ 7 4 a 7 + 1 1 ⋯ 1 7 7 2 ⋯ 4 8 a 6 − 1 1 ⋯ 1 7 6 0 ⋯ 5 5 a 5 + 1 8 ⋯ 1 7 5 3 ⋯ 7 3 a 4 − 2 1 ⋯ 2 3 2 4 ⋯ 4 6 a 3 + 2 4 ⋯ 4 7 1 6 ⋯ 8 8 a 2 − 6 7 ⋯ 0 1 2 8 ⋯ 9 7 a + 2 1 ⋯ 2 3 2 7 ⋯ 0 2 25 ⋯ 37 88 ⋯ 67 a 26 − 74 ⋯ 71 26 ⋯ 01 a 25 + 44 ⋯ 72 26 ⋯ 01 a 24 − 15 ⋯ 76 26 ⋯ 01 a 23 + 40 ⋯ 65 29 ⋯ 89 a 22 − 65 ⋯ 31 26 ⋯ 01 a 21 + 39 ⋯ 87 88 ⋯ 67 a 20 − 21 ⋯ 22 26 ⋯ 01 a 19 + 11 ⋯ 20 88 ⋯ 67 a 18 − 45 ⋯ 96 26 ⋯ 01 a 17 + 19 ⋯ 16 88 ⋯ 67 a 16 − 81 ⋯ 35 26 ⋯ 01 a 15 + 34 ⋯ 30 88 ⋯ 67 a 14 − 10 ⋯ 57 26 ⋯ 01 a 13 + 32 ⋯ 58 88 ⋯ 67 a 12 − 10 ⋯ 12 26 ⋯ 01 a 11 + 22 ⋯ 25 52 ⋯ 51 a 10 − 13 ⋯ 24 38 ⋯ 43 a 9 + 16 ⋯ 81 88 ⋯ 67 a 8 − 44 ⋯ 39 38 ⋯ 43 a 7 + 59 ⋯ 78 46 ⋯ 93 a 6 − 12 ⋯ 86 14 ⋯ 79 a 5 + 32 ⋯ 27 32 ⋯ 21 a 4 + 24 ⋯ 20 26 ⋯ 01 a 3 + 25 ⋯ 69 52 ⋯ 51 a 2 − 14 ⋯ 64 46 ⋯ 93 a + 47 ⋯ 00 11 ⋯ 61 \frac{25\cdots 37}{88\cdots 67}a^{26}-\frac{74\cdots 71}{26\cdots 01}a^{25}+\frac{44\cdots 72}{26\cdots 01}a^{24}-\frac{15\cdots 76}{26\cdots 01}a^{23}+\frac{40\cdots 65}{29\cdots 89}a^{22}-\frac{65\cdots 31}{26\cdots 01}a^{21}+\frac{39\cdots 87}{88\cdots 67}a^{20}-\frac{21\cdots 22}{26\cdots 01}a^{19}+\frac{11\cdots 20}{88\cdots 67}a^{18}-\frac{45\cdots 96}{26\cdots 01}a^{17}+\frac{19\cdots 16}{88\cdots 67}a^{16}-\frac{81\cdots 35}{26\cdots 01}a^{15}+\frac{34\cdots 30}{88\cdots 67}a^{14}-\frac{10\cdots 57}{26\cdots 01}a^{13}+\frac{32\cdots 58}{88\cdots 67}a^{12}-\frac{10\cdots 12}{26\cdots 01}a^{11}+\frac{22\cdots 25}{52\cdots 51}a^{10}-\frac{13\cdots 24}{38\cdots 43}a^{9}+\frac{16\cdots 81}{88\cdots 67}a^{8}-\frac{44\cdots 39}{38\cdots 43}a^{7}+\frac{59\cdots 78}{46\cdots 93}a^{6}-\frac{12\cdots 86}{14\cdots 79}a^{5}+\frac{32\cdots 27}{32\cdots 21}a^{4}+\frac{24\cdots 20}{26\cdots 01}a^{3}+\frac{25\cdots 69}{52\cdots 51}a^{2}-\frac{14\cdots 64}{46\cdots 93}a+\frac{47\cdots 00}{11\cdots 61} 8 8 ⋯ 6 7 2 5 ⋯ 3 7 a 2 6 − 2 6 ⋯ 0 1 7 4 ⋯ 7 1 a 2 5 + 2 6 ⋯ 0 1 4 4 ⋯ 7 2 a 2 4 − 2 6 ⋯ 0 1 1 5 ⋯ 7 6 a 2 3 + 2 9 ⋯ 8 9 4 0 ⋯ 6 5 a 2 2 − 2 6 ⋯ 0 1 6 5 ⋯ 3 1 a 2 1 + 8 8 ⋯ 6 7 3 9 ⋯ 8 7 a 2 0 − 2 6 ⋯ 0 1 2 1 ⋯ 2 2 a 1 9 + 8 8 ⋯ 6 7 1 1 ⋯ 2 0 a 1 8 − 2 6 ⋯ 0 1 4 5 ⋯ 9 6 a 1 7 + 8 8 ⋯ 6 7 1 9 ⋯ 1 6 a 1 6 − 2 6 ⋯ 0 1 8 1 ⋯ 3 5 a 1 5 + 8 8 ⋯ 6 7 3 4 ⋯ 3 0 a 1 4 − 2 6 ⋯ 0 1 1 0 ⋯ 5 7 a 1 3 + 8 8 ⋯ 6 7 3 2 ⋯ 5 8 a 1 2 − 2 6 ⋯ 0 1 1 0 ⋯ 1 2 a 1 1 + 5 2 ⋯ 5 1 2 2 ⋯ 2 5 a 1 0 − 3 8 ⋯ 4 3 1 3 ⋯ 2 4 a 9 + 8 8 ⋯ 6 7 1 6 ⋯ 8 1 a 8 − 3 8 ⋯ 4 3 4 4 ⋯ 3 9 a 7 + 4 6 ⋯ 9 3 5 9 ⋯ 7 8 a 6 − 1 4 ⋯ 7 9 1 2 ⋯ 8 6 a 5 + 3 2 ⋯ 2 1 3 2 ⋯ 2 7 a 4 + 2 6 ⋯ 0 1 2 4 ⋯ 2 0 a 3 + 5 2 ⋯ 5 1 2 5 ⋯ 6 9 a 2 − 4 6 ⋯ 9 3 1 4 ⋯ 6 4 a + 1 1 ⋯ 6 1 4 7 ⋯ 0 0 23 ⋯ 37 11 ⋯ 29 a 26 − 90 ⋯ 88 38 ⋯ 43 a 25 + 17 ⋯ 07 11 ⋯ 29 a 24 − 70 ⋯ 78 11 ⋯ 29 a 23 + 19 ⋯ 23 11 ⋯ 29 a 22 − 40 ⋯ 58 11 ⋯ 29 a 21 + 74 ⋯ 43 11 ⋯ 29 a 20 − 13 ⋯ 63 11 ⋯ 29 a 19 + 22 ⋯ 56 11 ⋯ 29 a 18 − 34 ⋯ 04 11 ⋯ 29 a 17 + 46 ⋯ 78 11 ⋯ 29 a 16 − 62 ⋯ 09 11 ⋯ 29 a 15 + 81 ⋯ 16 11 ⋯ 29 a 14 − 96 ⋯ 10 11 ⋯ 29 a 13 + 99 ⋯ 55 11 ⋯ 29 a 12 − 99 ⋯ 38 11 ⋯ 29 a 11 + 14 ⋯ 07 16 ⋯ 47 a 10 − 10 ⋯ 88 11 ⋯ 29 a 9 + 78 ⋯ 17 11 ⋯ 29 a 8 − 49 ⋯ 66 11 ⋯ 29 a 7 + 18 ⋯ 29 60 ⋯ 91 a 6 − 14 ⋯ 61 60 ⋯ 91 a 5 + 15 ⋯ 26 11 ⋯ 29 a 4 − 41 ⋯ 46 11 ⋯ 29 a 3 − 27 ⋯ 74 12 ⋯ 81 a 2 + 23 ⋯ 07 60 ⋯ 91 a − 25 ⋯ 71 11 ⋯ 29 \frac{23\cdots 37}{11\cdots 29}a^{26}-\frac{90\cdots 88}{38\cdots 43}a^{25}+\frac{17\cdots 07}{11\cdots 29}a^{24}-\frac{70\cdots 78}{11\cdots 29}a^{23}+\frac{19\cdots 23}{11\cdots 29}a^{22}-\frac{40\cdots 58}{11\cdots 29}a^{21}+\frac{74\cdots 43}{11\cdots 29}a^{20}-\frac{13\cdots 63}{11\cdots 29}a^{19}+\frac{22\cdots 56}{11\cdots 29}a^{18}-\frac{34\cdots 04}{11\cdots 29}a^{17}+\frac{46\cdots 78}{11\cdots 29}a^{16}-\frac{62\cdots 09}{11\cdots 29}a^{15}+\frac{81\cdots 16}{11\cdots 29}a^{14}-\frac{96\cdots 10}{11\cdots 29}a^{13}+\frac{99\cdots 55}{11\cdots 29}a^{12}-\frac{99\cdots 38}{11\cdots 29}a^{11}+\frac{14\cdots 07}{16\cdots 47}a^{10}-\frac{10\cdots 88}{11\cdots 29}a^{9}+\frac{78\cdots 17}{11\cdots 29}a^{8}-\frac{49\cdots 66}{11\cdots 29}a^{7}+\frac{18\cdots 29}{60\cdots 91}a^{6}-\frac{14\cdots 61}{60\cdots 91}a^{5}+\frac{15\cdots 26}{11\cdots 29}a^{4}-\frac{41\cdots 46}{11\cdots 29}a^{3}-\frac{27\cdots 74}{12\cdots 81}a^{2}+\frac{23\cdots 07}{60\cdots 91}a-\frac{25\cdots 71}{11\cdots 29} 1 1 ⋯ 2 9 2 3 ⋯ 3 7 a 2 6 − 3 8 ⋯ 4 3 9 0 ⋯ 8 8 a 2 5 + 1 1 ⋯ 2 9 1 7 ⋯ 0 7 a 2 4 − 1 1 ⋯ 2 9 7 0 ⋯ 7 8 a 2 3 + 1 1 ⋯ 2 9 1 9 ⋯ 2 3 a 2 2 − 1 1 ⋯ 2 9 4 0 ⋯ 5 8 a 2 1 + 1 1 ⋯ 2 9 7 4 ⋯ 4 3 a 2 0 − 1 1 ⋯ 2 9 1 3 ⋯ 6 3 a 1 9 + 1 1 ⋯ 2 9 2 2 ⋯ 5 6 a 1 8 − 1 1 ⋯ 2 9 3 4 ⋯ 0 4 a 1 7 + 1 1 ⋯ 2 9 4 6 ⋯ 7 8 a 1 6 − 1 1 ⋯ 2 9 6 2 ⋯ 0 9 a 1 5 + 1 1 ⋯ 2 9 8 1 ⋯ 1 6 a 1 4 − 1 1 ⋯ 2 9 9 6 ⋯ 1 0 a 1 3 + 1 1 ⋯ 2 9 9 9 ⋯ 5 5 a 1 2 − 1 1 ⋯ 2 9 9 9 ⋯ 3 8 a 1 1 + 1 6 ⋯ 4 7 1 4 ⋯ 0 7 a 1 0 − 1 1 ⋯ 2 9 1 0 ⋯ 8 8 a 9 + 1 1 ⋯ 2 9 7 8 ⋯ 1 7 a 8 − 1 1 ⋯ 2 9 4 9 ⋯ 6 6 a 7 + 6 0 ⋯ 9 1 1 8 ⋯ 2 9 a 6 − 6 0 ⋯ 9 1 1 4 ⋯ 6 1 a 5 + 1 1 ⋯ 2 9 1 5 ⋯ 2 6 a 4 − 1 1 ⋯ 2 9 4 1 ⋯ 4 6 a 3 − 1 2 ⋯ 8 1 2 7 ⋯ 7 4 a 2 + 6 0 ⋯ 9 1 2 3 ⋯ 0 7 a − 1 1 ⋯ 2 9 2 5 ⋯ 7 1 11 ⋯ 89 26 ⋯ 01 a 26 − 12 ⋯ 40 26 ⋯ 01 a 25 + 25 ⋯ 40 88 ⋯ 67 a 24 − 29 ⋯ 91 26 ⋯ 01 a 23 + 45 ⋯ 19 15 ⋯ 53 a 22 − 15 ⋯ 91 26 ⋯ 01 a 21 + 27 ⋯ 63 26 ⋯ 01 a 20 − 50 ⋯ 18 26 ⋯ 01 a 19 + 84 ⋯ 26 26 ⋯ 01 a 18 − 12 ⋯ 02 26 ⋯ 01 a 17 + 16 ⋯ 63 26 ⋯ 01 a 16 − 21 ⋯ 15 26 ⋯ 01 a 15 + 28 ⋯ 51 26 ⋯ 01 a 14 − 18 ⋯ 36 15 ⋯ 53 a 13 + 32 ⋯ 04 26 ⋯ 01 a 12 − 18 ⋯ 10 15 ⋯ 53 a 11 + 34 ⋯ 34 26 ⋯ 01 a 10 − 27 ⋯ 02 22 ⋯ 89 a 9 + 22 ⋯ 40 26 ⋯ 01 a 8 − 18 ⋯ 14 38 ⋯ 43 a 7 + 51 ⋯ 26 14 ⋯ 79 a 6 − 46 ⋯ 18 14 ⋯ 79 a 5 + 27 ⋯ 59 15 ⋯ 53 a 4 − 29 ⋯ 71 26 ⋯ 01 a 3 − 15 ⋯ 85 88 ⋯ 67 a 2 − 81 ⋯ 62 82 ⋯ 87 a + 25 ⋯ 00 26 ⋯ 01 \frac{11\cdots 89}{26\cdots 01}a^{26}-\frac{12\cdots 40}{26\cdots 01}a^{25}+\frac{25\cdots 40}{88\cdots 67}a^{24}-\frac{29\cdots 91}{26\cdots 01}a^{23}+\frac{45\cdots 19}{15\cdots 53}a^{22}-\frac{15\cdots 91}{26\cdots 01}a^{21}+\frac{27\cdots 63}{26\cdots 01}a^{20}-\frac{50\cdots 18}{26\cdots 01}a^{19}+\frac{84\cdots 26}{26\cdots 01}a^{18}-\frac{12\cdots 02}{26\cdots 01}a^{17}+\frac{16\cdots 63}{26\cdots 01}a^{16}-\frac{21\cdots 15}{26\cdots 01}a^{15}+\frac{28\cdots 51}{26\cdots 01}a^{14}-\frac{18\cdots 36}{15\cdots 53}a^{13}+\frac{32\cdots 04}{26\cdots 01}a^{12}-\frac{18\cdots 10}{15\cdots 53}a^{11}+\frac{34\cdots 34}{26\cdots 01}a^{10}-\frac{27\cdots 02}{22\cdots 89}a^{9}+\frac{22\cdots 40}{26\cdots 01}a^{8}-\frac{18\cdots 14}{38\cdots 43}a^{7}+\frac{51\cdots 26}{14\cdots 79}a^{6}-\frac{46\cdots 18}{14\cdots 79}a^{5}+\frac{27\cdots 59}{15\cdots 53}a^{4}-\frac{29\cdots 71}{26\cdots 01}a^{3}-\frac{15\cdots 85}{88\cdots 67}a^{2}-\frac{81\cdots 62}{82\cdots 87}a+\frac{25\cdots 00}{26\cdots 01} 2 6 ⋯ 0 1 1 1 ⋯ 8 9 a 2 6 − 2 6 ⋯ 0 1 1 2 ⋯ 4 0 a 2 5 + 8 8 ⋯ 6 7 2 5 ⋯ 4 0 a 2 4 − 2 6 ⋯ 0 1 2 9 ⋯ 9 1 a 2 3 + 1 5 ⋯ 5 3 4 5 ⋯ 1 9 a 2 2 − 2 6 ⋯ 0 1 1 5 ⋯ 9 1 a 2 1 + 2 6 ⋯ 0 1 2 7 ⋯ 6 3 a 2 0 − 2 6 ⋯ 0 1 5 0 ⋯ 1 8 a 1 9 + 2 6 ⋯ 0 1 8 4 ⋯ 2 6 a 1 8 − 2 6 ⋯ 0 1 1 2 ⋯ 0 2 a 1 7 + 2 6 ⋯ 0 1 1 6 ⋯ 6 3 a 1 6 − 2 6 ⋯ 0 1 2 1 ⋯ 1 5 a 1 5 + 2 6 ⋯ 0 1 2 8 ⋯ 5 1 a 1 4 − 1 5 ⋯ 5 3 1 8 ⋯ 3 6 a 1 3 + 2 6 ⋯ 0 1 3 2 ⋯ 0 4 a 1 2 − 1 5 ⋯ 5 3 1 8 ⋯ 1 0 a 1 1 + 2 6 ⋯ 0 1 3 4 ⋯ 3 4 a 1 0 − 2 2 ⋯ 8 9 2 7 ⋯ 0 2 a 9 + 2 6 ⋯ 0 1 2 2 ⋯ 4 0 a 8 − 3 8 ⋯ 4 3 1 8 ⋯ 1 4 a 7 + 1 4 ⋯ 7 9 5 1 ⋯ 2 6 a 6 − 1 4 ⋯ 7 9 4 6 ⋯ 1 8 a 5 + 1 5 ⋯ 5 3 2 7 ⋯ 5 9 a 4 − 2 6 ⋯ 0 1 2 9 ⋯ 7 1 a 3 − 8 8 ⋯ 6 7 1 5 ⋯ 8 5 a 2 − 8 2 ⋯ 8 7 8 1 ⋯ 6 2 a + 2 6 ⋯ 0 1 2 5 ⋯ 0 0 27 ⋯ 26 46 ⋯ 59 a 26 + 35 ⋯ 48 26 ⋯ 01 a 25 − 60 ⋯ 06 46 ⋯ 59 a 24 + 59 ⋯ 03 79 ⋯ 03 a 23 − 20 ⋯ 94 79 ⋯ 03 a 22 + 45 ⋯ 65 79 ⋯ 03 a 21 − 79 ⋯ 23 79 ⋯ 03 a 20 + 14 ⋯ 98 79 ⋯ 03 a 19 − 26 ⋯ 67 79 ⋯ 03 a 18 + 40 ⋯ 41 79 ⋯ 03 a 17 − 53 ⋯ 99 79 ⋯ 03 a 16 + 68 ⋯ 79 79 ⋯ 03 a 15 − 95 ⋯ 77 79 ⋯ 03 a 14 + 11 ⋯ 55 79 ⋯ 03 a 13 − 11 ⋯ 28 79 ⋯ 03 a 12 + 10 ⋯ 49 79 ⋯ 03 a 11 − 11 ⋯ 55 79 ⋯ 03 a 10 + 18 ⋯ 45 11 ⋯ 29 a 9 − 92 ⋯ 64 79 ⋯ 03 a 8 + 62 ⋯ 94 11 ⋯ 29 a 7 − 17 ⋯ 34 42 ⋯ 37 a 6 + 20 ⋯ 21 42 ⋯ 37 a 5 − 21 ⋯ 99 79 ⋯ 03 a 4 − 56 ⋯ 26 79 ⋯ 03 a 3 + 15 ⋯ 64 88 ⋯ 67 a 2 + 16 ⋯ 03 42 ⋯ 37 a + 11 ⋯ 44 79 ⋯ 03 \frac{27\cdots 26}{46\cdots 59}a^{26}+\frac{35\cdots 48}{26\cdots 01}a^{25}-\frac{60\cdots 06}{46\cdots 59}a^{24}+\frac{59\cdots 03}{79\cdots 03}a^{23}-\frac{20\cdots 94}{79\cdots 03}a^{22}+\frac{45\cdots 65}{79\cdots 03}a^{21}-\frac{79\cdots 23}{79\cdots 03}a^{20}+\frac{14\cdots 98}{79\cdots 03}a^{19}-\frac{26\cdots 67}{79\cdots 03}a^{18}+\frac{40\cdots 41}{79\cdots 03}a^{17}-\frac{53\cdots 99}{79\cdots 03}a^{16}+\frac{68\cdots 79}{79\cdots 03}a^{15}-\frac{95\cdots 77}{79\cdots 03}a^{14}+\frac{11\cdots 55}{79\cdots 03}a^{13}-\frac{11\cdots 28}{79\cdots 03}a^{12}+\frac{10\cdots 49}{79\cdots 03}a^{11}-\frac{11\cdots 55}{79\cdots 03}a^{10}+\frac{18\cdots 45}{11\cdots 29}a^{9}-\frac{92\cdots 64}{79\cdots 03}a^{8}+\frac{62\cdots 94}{11\cdots 29}a^{7}-\frac{17\cdots 34}{42\cdots 37}a^{6}+\frac{20\cdots 21}{42\cdots 37}a^{5}-\frac{21\cdots 99}{79\cdots 03}a^{4}-\frac{56\cdots 26}{79\cdots 03}a^{3}+\frac{15\cdots 64}{88\cdots 67}a^{2}+\frac{16\cdots 03}{42\cdots 37}a+\frac{11\cdots 44}{79\cdots 03} 4 6 ⋯ 5 9 2 7 ⋯ 2 6 a 2 6 + 2 6 ⋯ 0 1 3 5 ⋯ 4 8 a 2 5 − 4 6 ⋯ 5 9 6 0 ⋯ 0 6 a 2 4 + 7 9 ⋯ 0 3 5 9 ⋯ 0 3 a 2 3 − 7 9 ⋯ 0 3 2 0 ⋯ 9 4 a 2 2 + 7 9 ⋯ 0 3 4 5 ⋯ 6 5 a 2 1 − 7 9 ⋯ 0 3 7 9 ⋯ 2 3 a 2 0 + 7 9 ⋯ 0 3 1 4 ⋯ 9 8 a 1 9 − 7 9 ⋯ 0 3 2 6 ⋯ 6 7 a 1 8 + 7 9 ⋯ 0 3 4 0 ⋯ 4 1 a 1 7 − 7 9 ⋯ 0 3 5 3 ⋯ 9 9 a 1 6 + 7 9 ⋯ 0 3 6 8 ⋯ 7 9 a 1 5 − 7 9 ⋯ 0 3 9 5 ⋯ 7 7 a 1 4 + 7 9 ⋯ 0 3 1 1 ⋯ 5 5 a 1 3 − 7 9 ⋯ 0 3 1 1 ⋯ 2 8 a 1 2 + 7 9 ⋯ 0 3 1 0 ⋯ 4 9 a 1 1 − 7 9 ⋯ 0 3 1 1 ⋯ 5 5 a 1 0 + 1 1 ⋯ 2 9 1 8 ⋯ 4 5 a 9 − 7 9 ⋯ 0 3 9 2 ⋯ 6 4 a 8 + 1 1 ⋯ 2 9 6 2 ⋯ 9 4 a 7 − 4 2 ⋯ 3 7 1 7 ⋯ 3 4 a 6 + 4 2 ⋯ 3 7 2 0 ⋯ 2 1 a 5 − 7 9 ⋯ 0 3 2 1 ⋯ 9 9 a 4 − 7 9 ⋯ 0 3 5 6 ⋯ 2 6 a 3 + 8 8 ⋯ 6 7 1 5 ⋯ 6 4 a 2 + 4 2 ⋯ 3 7 1 6 ⋯ 0 3 a + 7 9 ⋯ 0 3 1 1 ⋯ 4 4 30 ⋯ 28 79 ⋯ 03 a 26 − 25 ⋯ 31 88 ⋯ 67 a 25 + 10 ⋯ 96 79 ⋯ 03 a 24 − 19 ⋯ 55 79 ⋯ 03 a 23 − 84 ⋯ 91 79 ⋯ 03 a 22 + 10 ⋯ 73 79 ⋯ 03 a 21 − 22 ⋯ 04 79 ⋯ 03 a 20 + 38 ⋯ 86 79 ⋯ 03 a 19 − 89 ⋯ 90 79 ⋯ 03 a 18 + 18 ⋯ 67 79 ⋯ 03 a 17 − 26 ⋯ 11 79 ⋯ 03 a 16 + 32 ⋯ 52 79 ⋯ 03 a 15 − 47 ⋯ 88 79 ⋯ 03 a 14 + 72 ⋯ 76 79 ⋯ 03 a 13 − 83 ⋯ 67 79 ⋯ 03 a 12 + 72 ⋯ 12 79 ⋯ 03 a 11 − 72 ⋯ 09 79 ⋯ 03 a 10 + 13 ⋯ 60 11 ⋯ 29 a 9 − 95 ⋯ 14 79 ⋯ 03 a 8 + 75 ⋯ 86 11 ⋯ 29 a 7 − 11 ⋯ 34 42 ⋯ 37 a 6 + 16 ⋯ 02 42 ⋯ 37 a 5 − 33 ⋯ 13 79 ⋯ 03 a 4 + 93 ⋯ 52 79 ⋯ 03 a 3 + 16 ⋯ 15 29 ⋯ 89 a 2 + 37 ⋯ 03 42 ⋯ 37 a − 24 ⋯ 03 79 ⋯ 03 \frac{30\cdots 28}{79\cdots 03}a^{26}-\frac{25\cdots 31}{88\cdots 67}a^{25}+\frac{10\cdots 96}{79\cdots 03}a^{24}-\frac{19\cdots 55}{79\cdots 03}a^{23}-\frac{84\cdots 91}{79\cdots 03}a^{22}+\frac{10\cdots 73}{79\cdots 03}a^{21}-\frac{22\cdots 04}{79\cdots 03}a^{20}+\frac{38\cdots 86}{79\cdots 03}a^{19}-\frac{89\cdots 90}{79\cdots 03}a^{18}+\frac{18\cdots 67}{79\cdots 03}a^{17}-\frac{26\cdots 11}{79\cdots 03}a^{16}+\frac{32\cdots 52}{79\cdots 03}a^{15}-\frac{47\cdots 88}{79\cdots 03}a^{14}+\frac{72\cdots 76}{79\cdots 03}a^{13}-\frac{83\cdots 67}{79\cdots 03}a^{12}+\frac{72\cdots 12}{79\cdots 03}a^{11}-\frac{72\cdots 09}{79\cdots 03}a^{10}+\frac{13\cdots 60}{11\cdots 29}a^{9}-\frac{95\cdots 14}{79\cdots 03}a^{8}+\frac{75\cdots 86}{11\cdots 29}a^{7}-\frac{11\cdots 34}{42\cdots 37}a^{6}+\frac{16\cdots 02}{42\cdots 37}a^{5}-\frac{33\cdots 13}{79\cdots 03}a^{4}+\frac{93\cdots 52}{79\cdots 03}a^{3}+\frac{16\cdots 15}{29\cdots 89}a^{2}+\frac{37\cdots 03}{42\cdots 37}a-\frac{24\cdots 03}{79\cdots 03} 7 9 ⋯ 0 3 3 0 ⋯ 2 8 a 2 6 − 8 8 ⋯ 6 7 2 5 ⋯ 3 1 a 2 5 + 7 9 ⋯ 0 3 1 0 ⋯ 9 6 a 2 4 − 7 9 ⋯ 0 3 1 9 ⋯ 5 5 a 2 3 − 7 9 ⋯ 0 3 8 4 ⋯ 9 1 a 2 2 + 7 9 ⋯ 0 3 1 0 ⋯ 7 3 a 2 1 − 7 9 ⋯ 0 3 2 2 ⋯ 0 4 a 2 0 + 7 9 ⋯ 0 3 3 8 ⋯ 8 6 a 1 9 − 7 9 ⋯ 0 3 8 9 ⋯ 9 0 a 1 8 + 7 9 ⋯ 0 3 1 8 ⋯ 6 7 a 1 7 − 7 9 ⋯ 0 3 2 6 ⋯ 1 1 a 1 6 + 7 9 ⋯ 0 3 3 2 ⋯ 5 2 a 1 5 − 7 9 ⋯ 0 3 4 7 ⋯ 8 8 a 1 4 + 7 9 ⋯ 0 3 7 2 ⋯ 7 6 a 1 3 − 7 9 ⋯ 0 3 8 3 ⋯ 6 7 a 1 2 + 7 9 ⋯ 0 3 7 2 ⋯ 1 2 a 1 1 − 7 9 ⋯ 0 3 7 2 ⋯ 0 9 a 1 0 + 1 1 ⋯ 2 9 1 3 ⋯ 6 0 a 9 − 7 9 ⋯ 0 3 9 5 ⋯ 1 4 a 8 + 1 1 ⋯ 2 9 7 5 ⋯ 8 6 a 7 − 4 2 ⋯ 3 7 1 1 ⋯ 3 4 a 6 + 4 2 ⋯ 3 7 1 6 ⋯ 0 2 a 5 − 7 9 ⋯ 0 3 3 3 ⋯ 1 3 a 4 + 7 9 ⋯ 0 3 9 3 ⋯ 5 2 a 3 + 2 9 ⋯ 8 9 1 6 ⋯ 1 5 a 2 + 4 2 ⋯ 3 7 3 7 ⋯ 0 3 a − 7 9 ⋯ 0 3 2 4 ⋯ 0 3
(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 : 35248883953.15641 35248883953.15641 3 5 2 4 8 8 8 3 9 5 3 . 1 5 6 4 1 (assuming GRH )
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
lim s → 1 ( s − 1 ) ζ K ( s ) = ( 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h w ⋅ ∣ D ∣ ≈ ( 2 1 ⋅ ( 2 π ) 13 ⋅ 35248883953.15641 ⋅ 5 2 ⋅ 8138911451501750747538217172562287688025999 ≈ ( 1.46950308511013
\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^{1}\cdot(2\pi)^{13}\cdot 35248883953.15641 \cdot 5}{2\cdot\sqrt{8138911451501750747538217172562287688025999}}\cr\approx \mathstrut & 1.46950308511013
\end{aligned} s → 1 lim ( s − 1 ) ζ K ( s ) = ( ≈ ( ≈ ( w ⋅ ∣ D ∣ 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h 2 ⋅ 8 1 3 8 9 1 1 4 5 1 5 0 1 7 5 0 7 4 7 5 3 8 2 1 7 1 7 2 5 6 2 2 8 7 6 8 8 0 2 5 9 9 9 2 1 ⋅ ( 2 π ) 1 3 ⋅ 3 5 2 4 8 8 8 3 9 5 3 . 1 5 6 4 1 ⋅ 5 1 . 4 6 9 5 0 3 0 8 5 1 1 0 1 3 (assuming GRH )
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1)
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))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1, 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)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1);
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)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1);
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))))
D 27 D_{27} D 2 7 27T8 ):
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)
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]
p p p 2 2 2 3 3 3 5 5 5 7 7 7 11 11 1 1 13 13 1 3 17 17 1 7 19 19 1 9 23 23 2 3 29 29 2 9 31 31 3 1 37 37 3 7 41 41 4 1 43 43 4 3 47 47 4 7 53 53 5 3 59 59 5 9
Cycle type
9 3 {\href{/padicField/2.9.0.1}{9} }^{3} 9 3 2 13 , 1 {\href{/padicField/3.2.0.1}{2} }^{13}{,}\,{\href{/padicField/3.1.0.1}{1} } 2 1 3 , 1 27 27 2 7 2 13 , 1 {\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} } 2 1 3 , 1 27 27 2 7 27 27 2 7 2 13 , 1 {\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} } 2 1 3 , 1 2 13 , 1 {\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} } 2 1 3 , 1 27 27 2 7 2 13 , 1 {\href{/padicField/29.2.0.1}{2} }^{13}{,}\,{\href{/padicField/29.1.0.1}{1} } 2 1 3 , 1 27 27 2 7 27 27 2 7 9 3 {\href{/padicField/41.9.0.1}{9} }^{3} 9 3 2 13 , 1 {\href{/padicField/43.2.0.1}{2} }^{13}{,}\,{\href{/padicField/43.1.0.1}{1} } 2 1 3 , 1 2 13 , 1 {\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} } 2 1 3 , 1 27 27 2 7 27 27 2 7
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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]
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
