sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1)
gp: K = bnfinit(y^18 + 2*y^16 - 5*y^15 - 3*y^14 + 22*y^13 - 23*y^12 + 45*y^11 + 117*y^10 - 54*y^9 - 54*y^8 - 21*y^7 + 15*y^6 + 39*y^5 - 63*y^4 + 17*y^3 + 15*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1)
x 18 + 2 x 16 − 5 x 15 − 3 x 14 + 22 x 13 − 23 x 12 + 45 x 11 + 117 x 10 − 54 x 9 + ⋯ + 1 x^{18} + 2 x^{16} - 5 x^{15} - 3 x^{14} + 22 x^{13} - 23 x^{12} + 45 x^{11} + 117 x^{10} - 54 x^{9} + \cdots + 1 x 1 8 + 2 x 1 6 − 5 x 1 5 − 3 x 1 4 + 2 2 x 1 3 − 2 3 x 1 2 + 4 5 x 1 1 + 1 1 7 x 1 0 − 5 4 x 9 + ⋯ + 1
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : 18 18 1 8
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : [ 6 , 6 ] [6, 6] [ 6 , 6 ]
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
24496374598915344039936 24496374598915344039936 2 4 4 9 6 3 7 4 5 9 8 9 1 5 3 4 4 0 3 9 9 3 6 = 2 18 ⋅ 3 9 ⋅ 7 15 \medspace = 2^{18}\cdot 3^{9}\cdot 7^{15} = 2 1 8 ⋅ 3 9 ⋅ 7 1 5
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : 17.53 17.53 1 7 . 5 3
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 : 2 3 / 2 3 1 / 2 7 5 / 6 ≈ 24.794421938893013 2^{3/2}3^{1/2}7^{5/6}\approx 24.794421938893013 2 3 / 2 3 1 / 2 7 5 / 6 ≈ 2 4 . 7 9 4 4 2 1 9 3 8 8 9 3 0 1 3 Ramified primes :
2 2 2 3 3 3 7 7 7
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : Q ( 21 ) \Q(\sqrt{21}) Q ( 2 1 )
Aut ( K / Q ) \Aut(K/\Q) A u t ( K / Q ) C 6 C_6 C 6
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is not Galois over Q \Q Q This is not a CM field .
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 a 7 − 1 3 a 6 + 1 3 a 4 − 1 3 a 3 + 1 3 a + 1 3 \frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3} 3 1 a 8 − 3 1 a 7 − 3 1 a 6 + 3 1 a 4 − 3 1 a 3 + 3 1 a + 3 1 1 3 a 9 + 1 3 a 7 − 1 3 a 6 + 1 3 a 5 − 1 3 a 3 + 1 3 a 2 − 1 3 a + 1 3 \frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3} 3 1 a 9 + 3 1 a 7 − 3 1 a 6 + 3 1 a 5 − 3 1 a 3 + 3 1 a 2 − 3 1 a + 3 1 1 3 a 10 − 1 3 a 6 + 1 3 a 4 − 1 3 a 3 − 1 3 a 2 − 1 3 \frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3} 3 1 a 1 0 − 3 1 a 6 + 3 1 a 4 − 3 1 a 3 − 3 1 a 2 − 3 1 1 3 a 11 − 1 3 a 7 + 1 3 a 5 − 1 3 a 4 − 1 3 a 3 − 1 3 a \frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a 3 1 a 1 1 − 3 1 a 7 + 3 1 a 5 − 3 1 a 4 − 3 1 a 3 − 3 1 a 1 3 a 12 − 1 3 a 7 − 1 3 a 5 − 1 3 a 3 − 1 3 a 2 + 1 3 a + 1 3 \frac{1}{3}a^{12}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3} 3 1 a 1 2 − 3 1 a 7 − 3 1 a 5 − 3 1 a 3 − 3 1 a 2 + 3 1 a + 3 1 1 3 a 13 − 1 3 a 7 + 1 3 a 6 + 1 3 a 3 + 1 3 a 2 − 1 3 a + 1 3 \frac{1}{3}a^{13}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3} 3 1 a 1 3 − 3 1 a 7 + 3 1 a 6 + 3 1 a 3 + 3 1 a 2 − 3 1 a + 3 1 1 3 a 14 − 1 3 a 6 − 1 3 a 4 − 1 3 a 2 − 1 3 a + 1 3 \frac{1}{3}a^{14}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3} 3 1 a 1 4 − 3 1 a 6 − 3 1 a 4 − 3 1 a 2 − 3 1 a + 3 1 1 15 a 15 + 1 15 a 12 + 2 15 a 11 − 1 15 a 9 + 2 15 a 8 − 1 15 a 7 − 4 15 a 6 + 1 3 a 5 + 2 5 a 4 + 1 15 a 3 − 1 5 a 2 + 2 5 a − 1 15 \frac{1}{15}a^{15}+\frac{1}{15}a^{12}+\frac{2}{15}a^{11}-\frac{1}{15}a^{9}+\frac{2}{15}a^{8}-\frac{1}{15}a^{7}-\frac{4}{15}a^{6}+\frac{1}{3}a^{5}+\frac{2}{5}a^{4}+\frac{1}{15}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{15} 1 5 1 a 1 5 + 1 5 1 a 1 2 + 1 5 2 a 1 1 − 1 5 1 a 9 + 1 5 2 a 8 − 1 5 1 a 7 − 1 5 4 a 6 + 3 1 a 5 + 5 2 a 4 + 1 5 1 a 3 − 5 1 a 2 + 5 2 a − 1 5 1 1 45 a 16 + 1 45 a 15 + 1 9 a 14 − 4 45 a 13 + 1 15 a 12 + 2 45 a 11 − 2 15 a 10 + 1 45 a 9 + 1 45 a 8 + 1 3 a 7 − 4 45 a 6 + 11 45 a 5 − 2 5 a 4 − 17 45 a 3 − 2 45 a 2 + 4 9 a + 4 45 \frac{1}{45}a^{16}+\frac{1}{45}a^{15}+\frac{1}{9}a^{14}-\frac{4}{45}a^{13}+\frac{1}{15}a^{12}+\frac{2}{45}a^{11}-\frac{2}{15}a^{10}+\frac{1}{45}a^{9}+\frac{1}{45}a^{8}+\frac{1}{3}a^{7}-\frac{4}{45}a^{6}+\frac{11}{45}a^{5}-\frac{2}{5}a^{4}-\frac{17}{45}a^{3}-\frac{2}{45}a^{2}+\frac{4}{9}a+\frac{4}{45} 4 5 1 a 1 6 + 4 5 1 a 1 5 + 9 1 a 1 4 − 4 5 4 a 1 3 + 1 5 1 a 1 2 + 4 5 2 a 1 1 − 1 5 2 a 1 0 + 4 5 1 a 9 + 4 5 1 a 8 + 3 1 a 7 − 4 5 4 a 6 + 4 5 1 1 a 5 − 5 2 a 4 − 4 5 1 7 a 3 − 4 5 2 a 2 + 9 4 a + 4 5 4 1 2115940095 a 17 − 5925712 2115940095 a 16 + 4092269 235104455 a 15 + 4842356 51608295 a 14 − 47617868 423188019 a 13 − 44599873 2115940095 a 12 + 288638171 2115940095 a 11 − 70675631 2115940095 a 10 + 348914594 2115940095 a 9 + 10156910 423188019 a 8 − 397719853 2115940095 a 7 + 294451567 2115940095 a 6 + 193197044 2115940095 a 5 + 757031191 2115940095 a 4 − 501480082 2115940095 a 3 + 137104358 705313365 a 2 − 19550264 705313365 a − 960084101 2115940095 \frac{1}{2115940095}a^{17}-\frac{5925712}{2115940095}a^{16}+\frac{4092269}{235104455}a^{15}+\frac{4842356}{51608295}a^{14}-\frac{47617868}{423188019}a^{13}-\frac{44599873}{2115940095}a^{12}+\frac{288638171}{2115940095}a^{11}-\frac{70675631}{2115940095}a^{10}+\frac{348914594}{2115940095}a^{9}+\frac{10156910}{423188019}a^{8}-\frac{397719853}{2115940095}a^{7}+\frac{294451567}{2115940095}a^{6}+\frac{193197044}{2115940095}a^{5}+\frac{757031191}{2115940095}a^{4}-\frac{501480082}{2115940095}a^{3}+\frac{137104358}{705313365}a^{2}-\frac{19550264}{705313365}a-\frac{960084101}{2115940095} 2 1 1 5 9 4 0 0 9 5 1 a 1 7 − 2 1 1 5 9 4 0 0 9 5 5 9 2 5 7 1 2 a 1 6 + 2 3 5 1 0 4 4 5 5 4 0 9 2 2 6 9 a 1 5 + 5 1 6 0 8 2 9 5 4 8 4 2 3 5 6 a 1 4 − 4 2 3 1 8 8 0 1 9 4 7 6 1 7 8 6 8 a 1 3 − 2 1 1 5 9 4 0 0 9 5 4 4 5 9 9 8 7 3 a 1 2 + 2 1 1 5 9 4 0 0 9 5 2 8 8 6 3 8 1 7 1 a 1 1 − 2 1 1 5 9 4 0 0 9 5 7 0 6 7 5 6 3 1 a 1 0 + 2 1 1 5 9 4 0 0 9 5 3 4 8 9 1 4 5 9 4 a 9 + 4 2 3 1 8 8 0 1 9 1 0 1 5 6 9 1 0 a 8 − 2 1 1 5 9 4 0 0 9 5 3 9 7 7 1 9 8 5 3 a 7 + 2 1 1 5 9 4 0 0 9 5 2 9 4 4 5 1 5 6 7 a 6 + 2 1 1 5 9 4 0 0 9 5 1 9 3 1 9 7 0 4 4 a 5 + 2 1 1 5 9 4 0 0 9 5 7 5 7 0 3 1 1 9 1 a 4 − 2 1 1 5 9 4 0 0 9 5 5 0 1 4 8 0 0 8 2 a 3 + 7 0 5 3 1 3 3 6 5 1 3 7 1 0 4 3 5 8 a 2 − 7 0 5 3 1 3 3 6 5 1 9 5 5 0 2 6 4 a − 2 1 1 5 9 4 0 0 9 5 9 6 0 0 8 4 1 0 1
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : Trivial group, which has order 1 1 1
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : C 2 C_{2} C 2 2 2 2
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 : 11 11 1 1
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 :
63744436 51608295 a 17 + 1235372 5734255 a 16 + 26773421 10321659 a 15 − 99072608 17202765 a 14 − 45684901 10321659 a 13 + 1331593691 51608295 a 12 − 1218088481 51608295 a 11 + 2710585009 51608295 a 10 + 521724959 3440553 a 9 − 1770948052 51608295 a 8 − 3486078202 51608295 a 7 − 647877538 17202765 a 6 + 652938574 51608295 a 5 + 482062310 10321659 a 4 − 1143869446 17202765 a 3 + 382829902 51608295 a 2 + 917938082 51608295 a − 54358373 10321659 \frac{63744436}{51608295}a^{17}+\frac{1235372}{5734255}a^{16}+\frac{26773421}{10321659}a^{15}-\frac{99072608}{17202765}a^{14}-\frac{45684901}{10321659}a^{13}+\frac{1331593691}{51608295}a^{12}-\frac{1218088481}{51608295}a^{11}+\frac{2710585009}{51608295}a^{10}+\frac{521724959}{3440553}a^{9}-\frac{1770948052}{51608295}a^{8}-\frac{3486078202}{51608295}a^{7}-\frac{647877538}{17202765}a^{6}+\frac{652938574}{51608295}a^{5}+\frac{482062310}{10321659}a^{4}-\frac{1143869446}{17202765}a^{3}+\frac{382829902}{51608295}a^{2}+\frac{917938082}{51608295}a-\frac{54358373}{10321659} 5 1 6 0 8 2 9 5 6 3 7 4 4 4 3 6 a 1 7 + 5 7 3 4 2 5 5 1 2 3 5 3 7 2 a 1 6 + 1 0 3 2 1 6 5 9 2 6 7 7 3 4 2 1 a 1 5 − 1 7 2 0 2 7 6 5 9 9 0 7 2 6 0 8 a 1 4 − 1 0 3 2 1 6 5 9 4 5 6 8 4 9 0 1 a 1 3 + 5 1 6 0 8 2 9 5 1 3 3 1 5 9 3 6 9 1 a 1 2 − 5 1 6 0 8 2 9 5 1 2 1 8 0 8 8 4 8 1 a 1 1 + 5 1 6 0 8 2 9 5 2 7 1 0 5 8 5 0 0 9 a 1 0 + 3 4 4 0 5 5 3 5 2 1 7 2 4 9 5 9 a 9 − 5 1 6 0 8 2 9 5 1 7 7 0 9 4 8 0 5 2 a 8 − 5 1 6 0 8 2 9 5 3 4 8 6 0 7 8 2 0 2 a 7 − 1 7 2 0 2 7 6 5 6 4 7 8 7 7 5 3 8 a 6 + 5 1 6 0 8 2 9 5 6 5 2 9 3 8 5 7 4 a 5 + 1 0 3 2 1 6 5 9 4 8 2 0 6 2 3 1 0 a 4 − 1 7 2 0 2 7 6 5 1 1 4 3 8 6 9 4 4 6 a 3 + 5 1 6 0 8 2 9 5 3 8 2 8 2 9 9 0 2 a 2 + 5 1 6 0 8 2 9 5 9 1 7 9 3 8 0 8 2 a − 1 0 3 2 1 6 5 9 5 4 3 5 8 3 7 3 8514302777 2115940095 a 17 + 664250977 705313365 a 16 + 17611346813 2115940095 a 15 − 104828857 5734255 a 14 − 6819100838 423188019 a 13 + 35573914241 423188019 a 12 − 152352187441 2115940095 a 11 + 347881373138 2115940095 a 10 + 357658464254 705313365 a 9 − 195966848513 2115940095 a 8 − 512606263247 2115940095 a 7 − 6808716326 47020891 a 6 + 57536633213 2115940095 a 5 + 337497017093 2115940095 a 4 − 148557897626 705313365 a 3 + 6462709912 423188019 a 2 + 136931556667 2115940095 a − 33450999773 2115940095 \frac{8514302777}{2115940095}a^{17}+\frac{664250977}{705313365}a^{16}+\frac{17611346813}{2115940095}a^{15}-\frac{104828857}{5734255}a^{14}-\frac{6819100838}{423188019}a^{13}+\frac{35573914241}{423188019}a^{12}-\frac{152352187441}{2115940095}a^{11}+\frac{347881373138}{2115940095}a^{10}+\frac{357658464254}{705313365}a^{9}-\frac{195966848513}{2115940095}a^{8}-\frac{512606263247}{2115940095}a^{7}-\frac{6808716326}{47020891}a^{6}+\frac{57536633213}{2115940095}a^{5}+\frac{337497017093}{2115940095}a^{4}-\frac{148557897626}{705313365}a^{3}+\frac{6462709912}{423188019}a^{2}+\frac{136931556667}{2115940095}a-\frac{33450999773}{2115940095} 2 1 1 5 9 4 0 0 9 5 8 5 1 4 3 0 2 7 7 7 a 1 7 + 7 0 5 3 1 3 3 6 5 6 6 4 2 5 0 9 7 7 a 1 6 + 2 1 1 5 9 4 0 0 9 5 1 7 6 1 1 3 4 6 8 1 3 a 1 5 − 5 7 3 4 2 5 5 1 0 4 8 2 8 8 5 7 a 1 4 − 4 2 3 1 8 8 0 1 9 6 8 1 9 1 0 0 8 3 8 a 1 3 + 4 2 3 1 8 8 0 1 9 3 5 5 7 3 9 1 4 2 4 1 a 1 2 − 2 1 1 5 9 4 0 0 9 5 1 5 2 3 5 2 1 8 7 4 4 1 a 1 1 + 2 1 1 5 9 4 0 0 9 5 3 4 7 8 8 1 3 7 3 1 3 8 a 1 0 + 7 0 5 3 1 3 3 6 5 3 5 7 6 5 8 4 6 4 2 5 4 a 9 − 2 1 1 5 9 4 0 0 9 5 1 9 5 9 6 6 8 4 8 5 1 3 a 8 − 2 1 1 5 9 4 0 0 9 5 5 1 2 6 0 6 2 6 3 2 4 7 a 7 − 4 7 0 2 0 8 9 1 6 8 0 8 7 1 6 3 2 6 a 6 + 2 1 1 5 9 4 0 0 9 5 5 7 5 3 6 6 3 3 2 1 3 a 5 + 2 1 1 5 9 4 0 0 9 5 3 3 7 4 9 7 0 1 7 0 9 3 a 4 − 7 0 5 3 1 3 3 6 5 1 4 8 5 5 7 8 9 7 6 2 6 a 3 + 4 2 3 1 8 8 0 1 9 6 4 6 2 7 0 9 9 1 2 a 2 + 2 1 1 5 9 4 0 0 9 5 1 3 6 9 3 1 5 5 6 6 6 7 a − 2 1 1 5 9 4 0 0 9 5 3 3 4 5 0 9 9 9 7 7 3 2591329 507785 a 17 + 5858929 4570065 a 16 + 48067759 4570065 a 15 − 2557994 111465 a 14 − 96324619 4570065 a 13 + 162685784 1523355 a 12 − 411338893 4570065 a 11 + 104928952 507785 a 10 + 2962723453 4570065 a 9 − 102135565 914013 a 8 − 94298950 304671 a 7 − 173365376 914013 a 6 + 26355598 914013 a 5 + 21137151 101557 a 4 − 1209266327 4570065 a 3 + 81716056 4570065 a 2 + 74578126 914013 a − 89649209 4570065 \frac{2591329}{507785}a^{17}+\frac{5858929}{4570065}a^{16}+\frac{48067759}{4570065}a^{15}-\frac{2557994}{111465}a^{14}-\frac{96324619}{4570065}a^{13}+\frac{162685784}{1523355}a^{12}-\frac{411338893}{4570065}a^{11}+\frac{104928952}{507785}a^{10}+\frac{2962723453}{4570065}a^{9}-\frac{102135565}{914013}a^{8}-\frac{94298950}{304671}a^{7}-\frac{173365376}{914013}a^{6}+\frac{26355598}{914013}a^{5}+\frac{21137151}{101557}a^{4}-\frac{1209266327}{4570065}a^{3}+\frac{81716056}{4570065}a^{2}+\frac{74578126}{914013}a-\frac{89649209}{4570065} 5 0 7 7 8 5 2 5 9 1 3 2 9 a 1 7 + 4 5 7 0 0 6 5 5 8 5 8 9 2 9 a 1 6 + 4 5 7 0 0 6 5 4 8 0 6 7 7 5 9 a 1 5 − 1 1 1 4 6 5 2 5 5 7 9 9 4 a 1 4 − 4 5 7 0 0 6 5 9 6 3 2 4 6 1 9 a 1 3 + 1 5 2 3 3 5 5 1 6 2 6 8 5 7 8 4 a 1 2 − 4 5 7 0 0 6 5 4 1 1 3 3 8 8 9 3 a 1 1 + 5 0 7 7 8 5 1 0 4 9 2 8 9 5 2 a 1 0 + 4 5 7 0 0 6 5 2 9 6 2 7 2 3 4 5 3 a 9 − 9 1 4 0 1 3 1 0 2 1 3 5 5 6 5 a 8 − 3 0 4 6 7 1 9 4 2 9 8 9 5 0 a 7 − 9 1 4 0 1 3 1 7 3 3 6 5 3 7 6 a 6 + 9 1 4 0 1 3 2 6 3 5 5 5 9 8 a 5 + 1 0 1 5 5 7 2 1 1 3 7 1 5 1 a 4 − 4 5 7 0 0 6 5 1 2 0 9 2 6 6 3 2 7 a 3 + 4 5 7 0 0 6 5 8 1 7 1 6 0 5 6 a 2 + 9 1 4 0 1 3 7 4 5 7 8 1 2 6 a − 4 5 7 0 0 6 5 8 9 6 4 9 2 0 9 1966926967 705313365 a 17 + 512300221 705313365 a 16 + 4040931836 705313365 a 15 − 215413963 17202765 a 14 − 1648673299 141062673 a 13 + 41091409958 705313365 a 12 − 2275790216 47020891 a 11 + 78915795923 705313365 a 10 + 83568282553 235104455 a 9 − 41119326127 705313365 a 8 − 24645137131 141062673 a 7 − 75567765832 705313365 a 6 + 10255383893 705313365 a 5 + 79558081181 705313365 a 4 − 6777283356 47020891 a 3 + 5539174526 705313365 a 2 + 6619739687 141062673 a − 2478614812 235104455 \frac{1966926967}{705313365}a^{17}+\frac{512300221}{705313365}a^{16}+\frac{4040931836}{705313365}a^{15}-\frac{215413963}{17202765}a^{14}-\frac{1648673299}{141062673}a^{13}+\frac{41091409958}{705313365}a^{12}-\frac{2275790216}{47020891}a^{11}+\frac{78915795923}{705313365}a^{10}+\frac{83568282553}{235104455}a^{9}-\frac{41119326127}{705313365}a^{8}-\frac{24645137131}{141062673}a^{7}-\frac{75567765832}{705313365}a^{6}+\frac{10255383893}{705313365}a^{5}+\frac{79558081181}{705313365}a^{4}-\frac{6777283356}{47020891}a^{3}+\frac{5539174526}{705313365}a^{2}+\frac{6619739687}{141062673}a-\frac{2478614812}{235104455} 7 0 5 3 1 3 3 6 5 1 9 6 6 9 2 6 9 6 7 a 1 7 + 7 0 5 3 1 3 3 6 5 5 1 2 3 0 0 2 2 1 a 1 6 + 7 0 5 3 1 3 3 6 5 4 0 4 0 9 3 1 8 3 6 a 1 5 − 1 7 2 0 2 7 6 5 2 1 5 4 1 3 9 6 3 a 1 4 − 1 4 1 0 6 2 6 7 3 1 6 4 8 6 7 3 2 9 9 a 1 3 + 7 0 5 3 1 3 3 6 5 4 1 0 9 1 4 0 9 9 5 8 a 1 2 − 4 7 0 2 0 8 9 1 2 2 7 5 7 9 0 2 1 6 a 1 1 + 7 0 5 3 1 3 3 6 5 7 8 9 1 5 7 9 5 9 2 3 a 1 0 + 2 3 5 1 0 4 4 5 5 8 3 5 6 8 2 8 2 5 5 3 a 9 − 7 0 5 3 1 3 3 6 5 4 1 1 1 9 3 2 6 1 2 7 a 8 − 1 4 1 0 6 2 6 7 3 2 4 6 4 5 1 3 7 1 3 1 a 7 − 7 0 5 3 1 3 3 6 5 7 5 5 6 7 7 6 5 8 3 2 a 6 + 7 0 5 3 1 3 3 6 5 1 0 2 5 5 3 8 3 8 9 3 a 5 + 7 0 5 3 1 3 3 6 5 7 9 5 5 8 0 8 1 1 8 1 a 4 − 4 7 0 2 0 8 9 1 6 7 7 7 2 8 3 3 5 6 a 3 + 7 0 5 3 1 3 3 6 5 5 5 3 9 1 7 4 5 2 6 a 2 + 1 4 1 0 6 2 6 7 3 6 6 1 9 7 3 9 6 8 7 a − 2 3 5 1 0 4 4 5 5 2 4 7 8 6 1 4 8 1 2 4475676353 705313365 a 17 + 457757016 235104455 a 16 + 3136213222 235104455 a 15 − 157916889 5734255 a 14 − 19297810531 705313365 a 13 + 92529491912 705313365 a 12 − 74982444016 705313365 a 11 + 178984076783 705313365 a 10 + 578977338572 705313365 a 9 − 63522136433 705313365 a 8 − 253756852193 705313365 a 7 − 166182774551 705313365 a 6 + 14882138831 705313365 a 5 + 35059490878 141062673 a 4 − 46373674532 141062673 a 3 + 7739926742 705313365 a 2 + 68859701458 705313365 a − 15360324626 705313365 \frac{4475676353}{705313365}a^{17}+\frac{457757016}{235104455}a^{16}+\frac{3136213222}{235104455}a^{15}-\frac{157916889}{5734255}a^{14}-\frac{19297810531}{705313365}a^{13}+\frac{92529491912}{705313365}a^{12}-\frac{74982444016}{705313365}a^{11}+\frac{178984076783}{705313365}a^{10}+\frac{578977338572}{705313365}a^{9}-\frac{63522136433}{705313365}a^{8}-\frac{253756852193}{705313365}a^{7}-\frac{166182774551}{705313365}a^{6}+\frac{14882138831}{705313365}a^{5}+\frac{35059490878}{141062673}a^{4}-\frac{46373674532}{141062673}a^{3}+\frac{7739926742}{705313365}a^{2}+\frac{68859701458}{705313365}a-\frac{15360324626}{705313365} 7 0 5 3 1 3 3 6 5 4 4 7 5 6 7 6 3 5 3 a 1 7 + 2 3 5 1 0 4 4 5 5 4 5 7 7 5 7 0 1 6 a 1 6 + 2 3 5 1 0 4 4 5 5 3 1 3 6 2 1 3 2 2 2 a 1 5 − 5 7 3 4 2 5 5 1 5 7 9 1 6 8 8 9 a 1 4 − 7 0 5 3 1 3 3 6 5 1 9 2 9 7 8 1 0 5 3 1 a 1 3 + 7 0 5 3 1 3 3 6 5 9 2 5 2 9 4 9 1 9 1 2 a 1 2 − 7 0 5 3 1 3 3 6 5 7 4 9 8 2 4 4 4 0 1 6 a 1 1 + 7 0 5 3 1 3 3 6 5 1 7 8 9 8 4 0 7 6 7 8 3 a 1 0 + 7 0 5 3 1 3 3 6 5 5 7 8 9 7 7 3 3 8 5 7 2 a 9 − 7 0 5 3 1 3 3 6 5 6 3 5 2 2 1 3 6 4 3 3 a 8 − 7 0 5 3 1 3 3 6 5 2 5 3 7 5 6 8 5 2 1 9 3 a 7 − 7 0 5 3 1 3 3 6 5 1 6 6 1 8 2 7 7 4 5 5 1 a 6 + 7 0 5 3 1 3 3 6 5 1 4 8 8 2 1 3 8 8 3 1 a 5 + 1 4 1 0 6 2 6 7 3 3 5 0 5 9 4 9 0 8 7 8 a 4 − 1 4 1 0 6 2 6 7 3 4 6 3 7 3 6 7 4 5 3 2 a 3 + 7 0 5 3 1 3 3 6 5 7 7 3 9 9 2 6 7 4 2 a 2 + 7 0 5 3 1 3 3 6 5 6 8 8 5 9 7 0 1 4 5 8 a − 7 0 5 3 1 3 3 6 5 1 5 3 6 0 3 2 4 6 2 6 3004478659 2115940095 a 17 + 308078111 2115940095 a 16 + 393263044 141062673 a 15 − 355855846 51608295 a 14 − 10914006106 2115940095 a 13 + 65304743897 2115940095 a 12 − 61463196814 2115940095 a 11 + 126886758937 2115940095 a 10 + 364239705848 2115940095 a 9 − 129146557487 2115940095 a 8 − 198198512404 2115940095 a 7 − 102026474336 2115940095 a 6 + 7684484272 423188019 a 5 + 128839115929 2115940095 a 4 − 172837065304 2115940095 a 3 + 11102837834 705313365 a 2 + 5650672446 235104455 a − 3152202202 423188019 \frac{3004478659}{2115940095}a^{17}+\frac{308078111}{2115940095}a^{16}+\frac{393263044}{141062673}a^{15}-\frac{355855846}{51608295}a^{14}-\frac{10914006106}{2115940095}a^{13}+\frac{65304743897}{2115940095}a^{12}-\frac{61463196814}{2115940095}a^{11}+\frac{126886758937}{2115940095}a^{10}+\frac{364239705848}{2115940095}a^{9}-\frac{129146557487}{2115940095}a^{8}-\frac{198198512404}{2115940095}a^{7}-\frac{102026474336}{2115940095}a^{6}+\frac{7684484272}{423188019}a^{5}+\frac{128839115929}{2115940095}a^{4}-\frac{172837065304}{2115940095}a^{3}+\frac{11102837834}{705313365}a^{2}+\frac{5650672446}{235104455}a-\frac{3152202202}{423188019} 2 1 1 5 9 4 0 0 9 5 3 0 0 4 4 7 8 6 5 9 a 1 7 + 2 1 1 5 9 4 0 0 9 5 3 0 8 0 7 8 1 1 1 a 1 6 + 1 4 1 0 6 2 6 7 3 3 9 3 2 6 3 0 4 4 a 1 5 − 5 1 6 0 8 2 9 5 3 5 5 8 5 5 8 4 6 a 1 4 − 2 1 1 5 9 4 0 0 9 5 1 0 9 1 4 0 0 6 1 0 6 a 1 3 + 2 1 1 5 9 4 0 0 9 5 6 5 3 0 4 7 4 3 8 9 7 a 1 2 − 2 1 1 5 9 4 0 0 9 5 6 1 4 6 3 1 9 6 8 1 4 a 1 1 + 2 1 1 5 9 4 0 0 9 5 1 2 6 8 8 6 7 5 8 9 3 7 a 1 0 + 2 1 1 5 9 4 0 0 9 5 3 6 4 2 3 9 7 0 5 8 4 8 a 9 − 2 1 1 5 9 4 0 0 9 5 1 2 9 1 4 6 5 5 7 4 8 7 a 8 − 2 1 1 5 9 4 0 0 9 5 1 9 8 1 9 8 5 1 2 4 0 4 a 7 − 2 1 1 5 9 4 0 0 9 5 1 0 2 0 2 6 4 7 4 3 3 6 a 6 + 4 2 3 1 8 8 0 1 9 7 6 8 4 4 8 4 2 7 2 a 5 + 2 1 1 5 9 4 0 0 9 5 1 2 8 8 3 9 1 1 5 9 2 9 a 4 − 2 1 1 5 9 4 0 0 9 5 1 7 2 8 3 7 0 6 5 3 0 4 a 3 + 7 0 5 3 1 3 3 6 5 1 1 1 0 2 8 3 7 8 3 4 a 2 + 2 3 5 1 0 4 4 5 5 5 6 5 0 6 7 2 4 4 6 a − 4 2 3 1 8 8 0 1 9 3 1 5 2 2 0 2 2 0 2 457757016 235104455 a 17 + 91457392 141062673 a 16 + 2954604418 705313365 a 15 − 143189792 17202765 a 14 − 1978462618 235104455 a 13 + 9319370701 235104455 a 12 − 22421359102 705313365 a 11 + 55323205271 705313365 a 10 + 178164386629 705313365 a 9 − 12070329131 705313365 a 8 − 72193571138 705313365 a 7 − 52253006464 705313365 a 6 + 746076623 705313365 a 5 + 50099237579 705313365 a 4 − 68346571259 705313365 a 3 + 1724556163 705313365 a 2 + 20445086198 705313365 a − 4475676353 705313365 \frac{457757016}{235104455}a^{17}+\frac{91457392}{141062673}a^{16}+\frac{2954604418}{705313365}a^{15}-\frac{143189792}{17202765}a^{14}-\frac{1978462618}{235104455}a^{13}+\frac{9319370701}{235104455}a^{12}-\frac{22421359102}{705313365}a^{11}+\frac{55323205271}{705313365}a^{10}+\frac{178164386629}{705313365}a^{9}-\frac{12070329131}{705313365}a^{8}-\frac{72193571138}{705313365}a^{7}-\frac{52253006464}{705313365}a^{6}+\frac{746076623}{705313365}a^{5}+\frac{50099237579}{705313365}a^{4}-\frac{68346571259}{705313365}a^{3}+\frac{1724556163}{705313365}a^{2}+\frac{20445086198}{705313365}a-\frac{4475676353}{705313365} 2 3 5 1 0 4 4 5 5 4 5 7 7 5 7 0 1 6 a 1 7 + 1 4 1 0 6 2 6 7 3 9 1 4 5 7 3 9 2 a 1 6 + 7 0 5 3 1 3 3 6 5 2 9 5 4 6 0 4 4 1 8 a 1 5 − 1 7 2 0 2 7 6 5 1 4 3 1 8 9 7 9 2 a 1 4 − 2 3 5 1 0 4 4 5 5 1 9 7 8 4 6 2 6 1 8 a 1 3 + 2 3 5 1 0 4 4 5 5 9 3 1 9 3 7 0 7 0 1 a 1 2 − 7 0 5 3 1 3 3 6 5 2 2 4 2 1 3 5 9 1 0 2 a 1 1 + 7 0 5 3 1 3 3 6 5 5 5 3 2 3 2 0 5 2 7 1 a 1 0 + 7 0 5 3 1 3 3 6 5 1 7 8 1 6 4 3 8 6 6 2 9 a 9 − 7 0 5 3 1 3 3 6 5 1 2 0 7 0 3 2 9 1 3 1 a 8 − 7 0 5 3 1 3 3 6 5 7 2 1 9 3 5 7 1 1 3 8 a 7 − 7 0 5 3 1 3 3 6 5 5 2 2 5 3 0 0 6 4 6 4 a 6 + 7 0 5 3 1 3 3 6 5 7 4 6 0 7 6 6 2 3 a 5 + 7 0 5 3 1 3 3 6 5 5 0 0 9 9 2 3 7 5 7 9 a 4 − 7 0 5 3 1 3 3 6 5 6 8 3 4 6 5 7 1 2 5 9 a 3 + 7 0 5 3 1 3 3 6 5 1 7 2 4 5 5 6 1 6 3 a 2 + 7 0 5 3 1 3 3 6 5 2 0 4 4 5 0 8 6 1 9 8 a − 7 0 5 3 1 3 3 6 5 4 4 7 5 6 7 6 3 5 3 759373754 235104455 a 17 + 1478577623 2115940095 a 16 + 2838464380 423188019 a 15 − 764257414 51608295 a 14 − 5308032946 423188019 a 13 + 47579377952 705313365 a 12 − 124615431056 2115940095 a 11 + 94230216118 705313365 a 10 + 170724236056 423188019 a 9 − 165899380057 2115940095 a 8 − 134476965794 705313365 a 7 − 240188508854 2115940095 a 6 + 56247184864 2115940095 a 5 + 17769560839 141062673 a 4 − 363042914218 2115940095 a 3 + 29102556092 2115940095 a 2 + 107009322217 2115940095 a − 5265341594 423188019 \frac{759373754}{235104455}a^{17}+\frac{1478577623}{2115940095}a^{16}+\frac{2838464380}{423188019}a^{15}-\frac{764257414}{51608295}a^{14}-\frac{5308032946}{423188019}a^{13}+\frac{47579377952}{705313365}a^{12}-\frac{124615431056}{2115940095}a^{11}+\frac{94230216118}{705313365}a^{10}+\frac{170724236056}{423188019}a^{9}-\frac{165899380057}{2115940095}a^{8}-\frac{134476965794}{705313365}a^{7}-\frac{240188508854}{2115940095}a^{6}+\frac{56247184864}{2115940095}a^{5}+\frac{17769560839}{141062673}a^{4}-\frac{363042914218}{2115940095}a^{3}+\frac{29102556092}{2115940095}a^{2}+\frac{107009322217}{2115940095}a-\frac{5265341594}{423188019} 2 3 5 1 0 4 4 5 5 7 5 9 3 7 3 7 5 4 a 1 7 + 2 1 1 5 9 4 0 0 9 5 1 4 7 8 5 7 7 6 2 3 a 1 6 + 4 2 3 1 8 8 0 1 9 2 8 3 8 4 6 4 3 8 0 a 1 5 − 5 1 6 0 8 2 9 5 7 6 4 2 5 7 4 1 4 a 1 4 − 4 2 3 1 8 8 0 1 9 5 3 0 8 0 3 2 9 4 6 a 1 3 + 7 0 5 3 1 3 3 6 5 4 7 5 7 9 3 7 7 9 5 2 a 1 2 − 2 1 1 5 9 4 0 0 9 5 1 2 4 6 1 5 4 3 1 0 5 6 a 1 1 + 7 0 5 3 1 3 3 6 5 9 4 2 3 0 2 1 6 1 1 8 a 1 0 + 4 2 3 1 8 8 0 1 9 1 7 0 7 2 4 2 3 6 0 5 6 a 9 − 2 1 1 5 9 4 0 0 9 5 1 6 5 8 9 9 3 8 0 0 5 7 a 8 − 7 0 5 3 1 3 3 6 5 1 3 4 4 7 6 9 6 5 7 9 4 a 7 − 2 1 1 5 9 4 0 0 9 5 2 4 0 1 8 8 5 0 8 8 5 4 a 6 + 2 1 1 5 9 4 0 0 9 5 5 6 2 4 7 1 8 4 8 6 4 a 5 + 1 4 1 0 6 2 6 7 3 1 7 7 6 9 5 6 0 8 3 9 a 4 − 2 1 1 5 9 4 0 0 9 5 3 6 3 0 4 2 9 1 4 2 1 8 a 3 + 2 1 1 5 9 4 0 0 9 5 2 9 1 0 2 5 5 6 0 9 2 a 2 + 2 1 1 5 9 4 0 0 9 5 1 0 7 0 0 9 3 2 2 2 1 7 a − 4 2 3 1 8 8 0 1 9 5 2 6 5 3 4 1 5 9 4 174232294 235104455 a 17 + 223941742 2115940095 a 16 + 3190997461 2115940095 a 15 − 181487669 51608295 a 14 − 5713075216 2115940095 a 13 + 742323469 47020891 a 12 − 31094932669 2115940095 a 11 + 4443427990 141062673 a 10 + 191313053707 2115940095 a 9 − 54460922204 2115940095 a 8 − 31312126193 705313365 a 7 − 52274753878 2115940095 a 6 + 15919511378 2115940095 a 5 + 3938817389 141062673 a 4 − 89090324039 2115940095 a 3 + 12565026523 2115940095 a 2 + 24188759849 2115940095 a − 8501373086 2115940095 \frac{174232294}{235104455}a^{17}+\frac{223941742}{2115940095}a^{16}+\frac{3190997461}{2115940095}a^{15}-\frac{181487669}{51608295}a^{14}-\frac{5713075216}{2115940095}a^{13}+\frac{742323469}{47020891}a^{12}-\frac{31094932669}{2115940095}a^{11}+\frac{4443427990}{141062673}a^{10}+\frac{191313053707}{2115940095}a^{9}-\frac{54460922204}{2115940095}a^{8}-\frac{31312126193}{705313365}a^{7}-\frac{52274753878}{2115940095}a^{6}+\frac{15919511378}{2115940095}a^{5}+\frac{3938817389}{141062673}a^{4}-\frac{89090324039}{2115940095}a^{3}+\frac{12565026523}{2115940095}a^{2}+\frac{24188759849}{2115940095}a-\frac{8501373086}{2115940095} 2 3 5 1 0 4 4 5 5 1 7 4 2 3 2 2 9 4 a 1 7 + 2 1 1 5 9 4 0 0 9 5 2 2 3 9 4 1 7 4 2 a 1 6 + 2 1 1 5 9 4 0 0 9 5 3 1 9 0 9 9 7 4 6 1 a 1 5 − 5 1 6 0 8 2 9 5 1 8 1 4 8 7 6 6 9 a 1 4 − 2 1 1 5 9 4 0 0 9 5 5 7 1 3 0 7 5 2 1 6 a 1 3 + 4 7 0 2 0 8 9 1 7 4 2 3 2 3 4 6 9 a 1 2 − 2 1 1 5 9 4 0 0 9 5 3 1 0 9 4 9 3 2 6 6 9 a 1 1 + 1 4 1 0 6 2 6 7 3 4 4 4 3 4 2 7 9 9 0 a 1 0 + 2 1 1 5 9 4 0 0 9 5 1 9 1 3 1 3 0 5 3 7 0 7 a 9 − 2 1 1 5 9 4 0 0 9 5 5 4 4 6 0 9 2 2 2 0 4 a 8 − 7 0 5 3 1 3 3 6 5 3 1 3 1 2 1 2 6 1 9 3 a 7 − 2 1 1 5 9 4 0 0 9 5 5 2 2 7 4 7 5 3 8 7 8 a 6 + 2 1 1 5 9 4 0 0 9 5 1 5 9 1 9 5 1 1 3 7 8 a 5 + 1 4 1 0 6 2 6 7 3 3 9 3 8 8 1 7 3 8 9 a 4 − 2 1 1 5 9 4 0 0 9 5 8 9 0 9 0 3 2 4 0 3 9 a 3 + 2 1 1 5 9 4 0 0 9 5 1 2 5 6 5 0 2 6 5 2 3 a 2 + 2 1 1 5 9 4 0 0 9 5 2 4 1 8 8 7 5 9 8 4 9 a − 2 1 1 5 9 4 0 0 9 5 8 5 0 1 3 7 3 0 8 6 16542466664 2115940095 a 17 + 4609842392 2115940095 a 16 + 34658062954 2115940095 a 15 − 1786111966 51608295 a 14 − 4609585619 141062673 a 13 + 342480834163 2115940095 a 12 − 94631736167 705313365 a 11 + 669188185436 2115940095 a 10 + 2114260810496 2115940095 a 9 − 18921582419 141062673 a 8 − 957721065422 2115940095 a 7 − 622317132362 2115940095 a 6 + 7738324209 235104455 a 5 + 651301739354 2115940095 a 4 − 848231312368 2115940095 a 3 + 40699174696 2115940095 a 2 + 255564984857 2115940095 a − 20245518083 705313365 \frac{16542466664}{2115940095}a^{17}+\frac{4609842392}{2115940095}a^{16}+\frac{34658062954}{2115940095}a^{15}-\frac{1786111966}{51608295}a^{14}-\frac{4609585619}{141062673}a^{13}+\frac{342480834163}{2115940095}a^{12}-\frac{94631736167}{705313365}a^{11}+\frac{669188185436}{2115940095}a^{10}+\frac{2114260810496}{2115940095}a^{9}-\frac{18921582419}{141062673}a^{8}-\frac{957721065422}{2115940095}a^{7}-\frac{622317132362}{2115940095}a^{6}+\frac{7738324209}{235104455}a^{5}+\frac{651301739354}{2115940095}a^{4}-\frac{848231312368}{2115940095}a^{3}+\frac{40699174696}{2115940095}a^{2}+\frac{255564984857}{2115940095}a-\frac{20245518083}{705313365} 2 1 1 5 9 4 0 0 9 5 1 6 5 4 2 4 6 6 6 6 4 a 1 7 + 2 1 1 5 9 4 0 0 9 5 4 6 0 9 8 4 2 3 9 2 a 1 6 + 2 1 1 5 9 4 0 0 9 5 3 4 6 5 8 0 6 2 9 5 4 a 1 5 − 5 1 6 0 8 2 9 5 1 7 8 6 1 1 1 9 6 6 a 1 4 − 1 4 1 0 6 2 6 7 3 4 6 0 9 5 8 5 6 1 9 a 1 3 + 2 1 1 5 9 4 0 0 9 5 3 4 2 4 8 0 8 3 4 1 6 3 a 1 2 − 7 0 5 3 1 3 3 6 5 9 4 6 3 1 7 3 6 1 6 7 a 1 1 + 2 1 1 5 9 4 0 0 9 5 6 6 9 1 8 8 1 8 5 4 3 6 a 1 0 + 2 1 1 5 9 4 0 0 9 5 2 1 1 4 2 6 0 8 1 0 4 9 6 a 9 − 1 4 1 0 6 2 6 7 3 1 8 9 2 1 5 8 2 4 1 9 a 8 − 2 1 1 5 9 4 0 0 9 5 9 5 7 7 2 1 0 6 5 4 2 2 a 7 − 2 1 1 5 9 4 0 0 9 5 6 2 2 3 1 7 1 3 2 3 6 2 a 6 + 2 3 5 1 0 4 4 5 5 7 7 3 8 3 2 4 2 0 9 a 5 + 2 1 1 5 9 4 0 0 9 5 6 5 1 3 0 1 7 3 9 3 5 4 a 4 − 2 1 1 5 9 4 0 0 9 5 8 4 8 2 3 1 3 1 2 3 6 8 a 3 + 2 1 1 5 9 4 0 0 9 5 4 0 6 9 9 1 7 4 6 9 6 a 2 + 2 1 1 5 9 4 0 0 9 5 2 5 5 5 6 4 9 8 4 8 5 7 a − 7 0 5 3 1 3 3 6 5 2 0 2 4 5 5 1 8 0 8 3 9669940978 2115940095 a 17 + 1117519036 705313365 a 16 + 20426170933 2115940095 a 15 − 334521904 17202765 a 14 − 43521937631 2115940095 a 13 + 198422713694 2115940095 a 12 − 154362867647 2115940095 a 11 + 380602681138 2115940095 a 10 + 28147671841 47020891 a 9 − 90004421659 2115940095 a 8 − 110626081418 423188019 a 7 − 128553006893 705313365 a 6 + 10641263491 2115940095 a 5 + 382456164847 2115940095 a 4 − 160937670961 705313365 a 3 − 20904104 2115940095 a 2 + 29894962549 423188019 a − 28863720118 2115940095 \frac{9669940978}{2115940095}a^{17}+\frac{1117519036}{705313365}a^{16}+\frac{20426170933}{2115940095}a^{15}-\frac{334521904}{17202765}a^{14}-\frac{43521937631}{2115940095}a^{13}+\frac{198422713694}{2115940095}a^{12}-\frac{154362867647}{2115940095}a^{11}+\frac{380602681138}{2115940095}a^{10}+\frac{28147671841}{47020891}a^{9}-\frac{90004421659}{2115940095}a^{8}-\frac{110626081418}{423188019}a^{7}-\frac{128553006893}{705313365}a^{6}+\frac{10641263491}{2115940095}a^{5}+\frac{382456164847}{2115940095}a^{4}-\frac{160937670961}{705313365}a^{3}-\frac{20904104}{2115940095}a^{2}+\frac{29894962549}{423188019}a-\frac{28863720118}{2115940095} 2 1 1 5 9 4 0 0 9 5 9 6 6 9 9 4 0 9 7 8 a 1 7 + 7 0 5 3 1 3 3 6 5 1 1 1 7 5 1 9 0 3 6 a 1 6 + 2 1 1 5 9 4 0 0 9 5 2 0 4 2 6 1 7 0 9 3 3 a 1 5 − 1 7 2 0 2 7 6 5 3 3 4 5 2 1 9 0 4 a 1 4 − 2 1 1 5 9 4 0 0 9 5 4 3 5 2 1 9 3 7 6 3 1 a 1 3 + 2 1 1 5 9 4 0 0 9 5 1 9 8 4 2 2 7 1 3 6 9 4 a 1 2 − 2 1 1 5 9 4 0 0 9 5 1 5 4 3 6 2 8 6 7 6 4 7 a 1 1 + 2 1 1 5 9 4 0 0 9 5 3 8 0 6 0 2 6 8 1 1 3 8 a 1 0 + 4 7 0 2 0 8 9 1 2 8 1 4 7 6 7 1 8 4 1 a 9 − 2 1 1 5 9 4 0 0 9 5 9 0 0 0 4 4 2 1 6 5 9 a 8 − 4 2 3 1 8 8 0 1 9 1 1 0 6 2 6 0 8 1 4 1 8 a 7 − 7 0 5 3 1 3 3 6 5 1 2 8 5 5 3 0 0 6 8 9 3 a 6 + 2 1 1 5 9 4 0 0 9 5 1 0 6 4 1 2 6 3 4 9 1 a 5 + 2 1 1 5 9 4 0 0 9 5 3 8 2 4 5 6 1 6 4 8 4 7 a 4 − 7 0 5 3 1 3 3 6 5 1 6 0 9 3 7 6 7 0 9 6 1 a 3 − 2 1 1 5 9 4 0 0 9 5 2 0 9 0 4 1 0 4 a 2 + 4 2 3 1 8 8 0 1 9 2 9 8 9 4 9 6 2 5 4 9 a − 2 1 1 5 9 4 0 0 9 5 2 8 8 6 3 7 2 0 1 1 8
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 : 14420.859848954282 14420.859848954282 1 4 4 2 0 . 8 5 9 8 4 8 9 5 4 2 8 2
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 6 ⋅ ( 2 π ) 6 ⋅ 14420.859848954282 ⋅ 1 2 ⋅ 24496374598915344039936 ≈ ( 0.181413431094210
\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^{6}\cdot(2\pi)^{6}\cdot 14420.859848954282 \cdot 1}{2\cdot\sqrt{24496374598915344039936}}\cr\approx \mathstrut & 0.181413431094210
\end{aligned} s → 1 lim ( s − 1 ) ζ K ( s ) = ( ≈ ( ≈ ( w ⋅ ∣ D ∣ 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h 2 ⋅ 2 4 4 9 6 3 7 4 5 9 8 9 1 5 3 4 4 0 3 9 9 3 6 2 6 ⋅ ( 2 π ) 6 ⋅ 1 4 4 2 0 . 8 5 9 8 4 8 9 5 4 2 8 2 ⋅ 1 0 . 1 8 1 4 1 3 4 3 1 0 9 4 2 1 0
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*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^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*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^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*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^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*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))))
C 6 × S 3 C_6\times S_3 C 6 × S 3 18T6 ):
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
R
R
3 6 {\href{/padicField/5.3.0.1}{3} }^{6} 3 6 R
6 3 {\href{/padicField/11.6.0.1}{6} }^{3} 6 3 2 9 {\href{/padicField/13.2.0.1}{2} }^{9} 2 9 6 2 , 3 2 {\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2} 6 2 , 3 2 6 3 {\href{/padicField/19.6.0.1}{6} }^{3} 6 3 6 3 {\href{/padicField/23.6.0.1}{6} }^{3} 6 3 6 3 {\href{/padicField/29.6.0.1}{6} }^{3} 6 3 6 3 {\href{/padicField/31.6.0.1}{6} }^{3} 6 3 6 2 , 3 2 {\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2} 6 2 , 3 2 2 6 , 1 6 {\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6} 2 6 , 1 6 2 6 , 1 6 {\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6} 2 6 , 1 6 6 2 , 3 2 {\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2} 6 2 , 3 2 6 3 {\href{/padicField/53.6.0.1}{6} }^{3} 6 3 3 6 {\href{/padicField/59.3.0.1}{3} }^{6} 3 6
In the table, R denotes a ramified prime.
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]
p p p Label Polynomial
e e e f f f c c c Galois group Slope content
2 2 2
2.6.1.0a1.1 x 6 + x 4 + x 3 + x + 1 x^{6} + x^{4} + x^{3} + x + 1 x 6 + x 4 + x 3 + x + 1 1 1 1 6 6 6 0 0 0 C 6 C_6 C 6 [ ] 6 [\ ]^{6} [ ] 6 2.6.2.18a1.17 x 12 + 2 x 10 + 2 x 9 + x 8 + 4 x 7 + 7 x 6 + 2 x 5 + 8 x 4 + 6 x 3 + x 2 + 6 x + 7 x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 7 x^{6} + 2 x^{5} + 8 x^{4} + 6 x^{3} + x^{2} + 6 x + 7 x 1 2 + 2 x 1 0 + 2 x 9 + x 8 + 4 x 7 + 7 x 6 + 2 x 5 + 8 x 4 + 6 x 3 + x 2 + 6 x + 7 2 2 2 6 6 6 18 18 1 8 C 6 × C 2 C_6\times C_2 C 6 × C 2 [ 3 ] 6 [3]^{6} [ 3 ] 6
3 3 3
3.3.2.3a1.1 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 7 x + 1 x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 7 x + 1 2 2 2 3 3 3 3 3 3 C 6 C_6 C 6 [ ] 2 3 [\ ]_{2}^{3} [ ] 2 3 3.3.2.3a1.1 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 7 x + 1 x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 7 x + 1 2 2 2 3 3 3 3 3 3 C 6 C_6 C 6 [ ] 2 3 [\ ]_{2}^{3} [ ] 2 3 3.3.2.3a1.1 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 7 x + 1 x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 7 x + 1 2 2 2 3 3 3 3 3 3 C 6 C_6 C 6 [ ] 2 3 [\ ]_{2}^{3} [ ] 2 3
7 7 7
7.3.6.15a1.3 x 18 + 36 x 17 + 540 x 16 + 4344 x 15 + 20160 x 14 + 55296 x 13 + 98736 x 12 + 161280 x 11 + 238464 x 10 + 208640 x 9 + 334080 x 8 + 138240 x 7 + 280320 x 6 + 46080 x 5 + 138240 x 4 + 6144 x 3 + 36864 x 2 + 4103 x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98736 x^{12} + 161280 x^{11} + 238464 x^{10} + 208640 x^{9} + 334080 x^{8} + 138240 x^{7} + 280320 x^{6} + 46080 x^{5} + 138240 x^{4} + 6144 x^{3} + 36864 x^{2} + 4103 x 1 8 + 3 6 x 1 7 + 5 4 0 x 1 6 + 4 3 4 4 x 1 5 + 2 0 1 6 0 x 1 4 + 5 5 2 9 6 x 1 3 + 9 8 7 3 6 x 1 2 + 1 6 1 2 8 0 x 1 1 + 2 3 8 4 6 4 x 1 0 + 2 0 8 6 4 0 x 9 + 3 3 4 0 8 0 x 8 + 1 3 8 2 4 0 x 7 + 2 8 0 3 2 0 x 6 + 4 6 0 8 0 x 5 + 1 3 8 2 4 0 x 4 + 6 1 4 4 x 3 + 3 6 8 6 4 x 2 + 4 1 0 3 6 6 6 3 3 3 15 15 1 5 C 6 × C 3 C_6 \times C_3 C 6 × C 3 [ ] 6 3 [\ ]_{6}^{3} [ ] 6 3
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
