Properties

 Label 1.25.20t1.a Dimension $1$ Group $C_{20}$ Conductor $25$ Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: 20T1 Conductor: $$25$$$$\medspace = 5^{2}$$ Artin number field: Galois closure of $$\Q(\zeta_{25})$$ Galois orbit size: $8$ Smallest permutation container: 20T1 Parity: odd Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{10} + x^{6} + 25x^{5} + 8x^{4} + 17x^{3} + 2x^{2} + 22x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$14 a^{9} + 24 a^{8} + 21 a^{7} + 19 a^{6} + 26 a^{5} + 18 a^{3} + 20 a^{2} + 7 a + 22 + \left(21 a^{9} + 6 a^{8} + 12 a^{7} + 24 a^{6} + 15 a^{5} + 14 a^{4} + 17 a^{3} + 24 a^{2} + 2 a + 3\right)\cdot 29 + \left(11 a^{9} + 24 a^{8} + 19 a^{7} + 14 a^{6} + 18 a^{5} + 26 a^{4} + 14 a^{3} + a^{2} + 3 a\right)\cdot 29^{2} + \left(26 a^{9} + 2 a^{8} + 27 a^{7} + 22 a^{6} + 2 a^{5} + 2 a^{4} + 6 a^{3} + 15 a^{2} + 22 a + 15\right)\cdot 29^{3} + \left(15 a^{9} + 4 a^{8} + 21 a^{7} + 26 a^{6} + 12 a^{5} + 7 a^{4} + 2 a^{3} + 7 a^{2} + 20 a + 18\right)\cdot 29^{4} +O(29^{5})$$ 14*a^9 + 24*a^8 + 21*a^7 + 19*a^6 + 26*a^5 + 18*a^3 + 20*a^2 + 7*a + 22 + (21*a^9 + 6*a^8 + 12*a^7 + 24*a^6 + 15*a^5 + 14*a^4 + 17*a^3 + 24*a^2 + 2*a + 3)*29 + (11*a^9 + 24*a^8 + 19*a^7 + 14*a^6 + 18*a^5 + 26*a^4 + 14*a^3 + a^2 + 3*a)*29^2 + (26*a^9 + 2*a^8 + 27*a^7 + 22*a^6 + 2*a^5 + 2*a^4 + 6*a^3 + 15*a^2 + 22*a + 15)*29^3 + (15*a^9 + 4*a^8 + 21*a^7 + 26*a^6 + 12*a^5 + 7*a^4 + 2*a^3 + 7*a^2 + 20*a + 18)*29^4+O(29^5) $r_{ 2 }$ $=$ $$24 a^{9} + 28 a^{8} + 14 a^{7} + 21 a^{6} + 8 a^{5} + 4 a^{4} + a^{3} + 3 a^{2} + 27 a + 13 + \left(26 a^{9} + 5 a^{8} + 2 a^{7} + 27 a^{6} + 23 a^{5} + 24 a^{4} + 10 a^{3} + 9 a^{2} + 24 a + 6\right)\cdot 29 + \left(25 a^{9} + 9 a^{8} + 16 a^{7} + 20 a^{6} + 4 a^{5} + 23 a^{4} + 20 a^{3} + 27 a^{2} + 13 a + 24\right)\cdot 29^{2} + \left(11 a^{9} + 19 a^{8} + 9 a^{7} + 13 a^{6} + 5 a^{5} + 11 a^{4} + 3 a^{3} + 4 a^{2} + 7 a + 10\right)\cdot 29^{3} + \left(22 a^{9} + 14 a^{8} + 14 a^{7} + 16 a^{6} + 23 a^{5} + 20 a^{4} + a^{3} + 4 a^{2} + 12 a + 17\right)\cdot 29^{4} +O(29^{5})$$ 24*a^9 + 28*a^8 + 14*a^7 + 21*a^6 + 8*a^5 + 4*a^4 + a^3 + 3*a^2 + 27*a + 13 + (26*a^9 + 5*a^8 + 2*a^7 + 27*a^6 + 23*a^5 + 24*a^4 + 10*a^3 + 9*a^2 + 24*a + 6)*29 + (25*a^9 + 9*a^8 + 16*a^7 + 20*a^6 + 4*a^5 + 23*a^4 + 20*a^3 + 27*a^2 + 13*a + 24)*29^2 + (11*a^9 + 19*a^8 + 9*a^7 + 13*a^6 + 5*a^5 + 11*a^4 + 3*a^3 + 4*a^2 + 7*a + 10)*29^3 + (22*a^9 + 14*a^8 + 14*a^7 + 16*a^6 + 23*a^5 + 20*a^4 + a^3 + 4*a^2 + 12*a + 17)*29^4+O(29^5) $r_{ 3 }$ $=$ $$13 a^{9} + 22 a^{8} + 22 a^{7} + 6 a^{6} + 8 a^{4} + 6 a^{3} + 17 a + 27 + \left(28 a^{9} + 26 a^{8} + 11 a^{7} + 23 a^{6} + 10 a^{5} + 25 a^{3} + 4 a^{2} + 2 a + 6\right)\cdot 29 + \left(11 a^{8} + a^{7} + 4 a^{6} + 26 a^{5} + 17 a^{4} + 19 a^{2} + 8 a + 19\right)\cdot 29^{2} + \left(15 a^{9} + 19 a^{8} + 27 a^{7} + 6 a^{6} + 4 a^{5} + 2 a^{3} + 6 a^{2} + 14 a + 21\right)\cdot 29^{3} + \left(16 a^{9} + 6 a^{8} + 5 a^{7} + 10 a^{6} + 13 a^{5} + 5 a^{4} + 28 a^{3} + 6 a^{2} + 27 a + 14\right)\cdot 29^{4} +O(29^{5})$$ 13*a^9 + 22*a^8 + 22*a^7 + 6*a^6 + 8*a^4 + 6*a^3 + 17*a + 27 + (28*a^9 + 26*a^8 + 11*a^7 + 23*a^6 + 10*a^5 + 25*a^3 + 4*a^2 + 2*a + 6)*29 + (11*a^8 + a^7 + 4*a^6 + 26*a^5 + 17*a^4 + 19*a^2 + 8*a + 19)*29^2 + (15*a^9 + 19*a^8 + 27*a^7 + 6*a^6 + 4*a^5 + 2*a^3 + 6*a^2 + 14*a + 21)*29^3 + (16*a^9 + 6*a^8 + 5*a^7 + 10*a^6 + 13*a^5 + 5*a^4 + 28*a^3 + 6*a^2 + 27*a + 14)*29^4+O(29^5) $r_{ 4 }$ $=$ $$4 a^{9} + 7 a^{8} + 7 a^{7} + 21 a^{6} + 20 a^{5} + 26 a^{4} + 6 a^{3} + 17 a^{2} + 15 a + 10 + \left(8 a^{9} + 4 a^{8} + 20 a^{7} + 11 a^{6} + 6 a^{5} + 8 a^{4} + 6 a^{2} + 14 a + 22\right)\cdot 29 + \left(5 a^{9} + 15 a^{8} + 18 a^{7} + 7 a^{6} + 4 a^{5} + 22 a^{3} + 23 a^{2} + 9 a + 1\right)\cdot 29^{2} + \left(16 a^{9} + 11 a^{8} + 7 a^{7} + 3 a^{6} + 9 a^{5} + 24 a^{4} + 9 a^{3} + 20 a^{2} + 25 a + 15\right)\cdot 29^{3} + \left(10 a^{9} + 4 a^{8} + 20 a^{7} + 26 a^{6} + 5 a^{5} + 9 a^{4} + 19 a^{3} + 5 a^{2} + 24 a + 10\right)\cdot 29^{4} +O(29^{5})$$ 4*a^9 + 7*a^8 + 7*a^7 + 21*a^6 + 20*a^5 + 26*a^4 + 6*a^3 + 17*a^2 + 15*a + 10 + (8*a^9 + 4*a^8 + 20*a^7 + 11*a^6 + 6*a^5 + 8*a^4 + 6*a^2 + 14*a + 22)*29 + (5*a^9 + 15*a^8 + 18*a^7 + 7*a^6 + 4*a^5 + 22*a^3 + 23*a^2 + 9*a + 1)*29^2 + (16*a^9 + 11*a^8 + 7*a^7 + 3*a^6 + 9*a^5 + 24*a^4 + 9*a^3 + 20*a^2 + 25*a + 15)*29^3 + (10*a^9 + 4*a^8 + 20*a^7 + 26*a^6 + 5*a^5 + 9*a^4 + 19*a^3 + 5*a^2 + 24*a + 10)*29^4+O(29^5) $r_{ 5 }$ $=$ $$16 a^{9} + 12 a^{8} + 15 a^{7} + 27 a^{6} + 23 a^{5} + 26 a^{4} + 17 a^{2} + 3 + \left(21 a^{9} + 16 a^{8} + 6 a^{7} + 17 a^{6} + 2 a^{5} + 21 a^{4} + 4 a^{3} + 3 a^{2} + 26 a + 15\right)\cdot 29 + \left(3 a^{8} + 4 a^{7} + 28 a^{6} + 5 a^{5} + 23 a^{4} + 6 a^{3} + 13 a^{2} + 3 a + 19\right)\cdot 29^{2} + \left(17 a^{9} + 19 a^{8} + 13 a^{7} + 4 a^{6} + 18 a^{5} + 17 a^{4} + 9 a^{3} + 13 a^{2} + 3 a + 14\right)\cdot 29^{3} + \left(26 a^{9} + 3 a^{8} + 22 a^{7} + 11 a^{6} + 8 a^{4} + 15 a^{3} + 16 a^{2} + 23 a + 9\right)\cdot 29^{4} +O(29^{5})$$ 16*a^9 + 12*a^8 + 15*a^7 + 27*a^6 + 23*a^5 + 26*a^4 + 17*a^2 + 3 + (21*a^9 + 16*a^8 + 6*a^7 + 17*a^6 + 2*a^5 + 21*a^4 + 4*a^3 + 3*a^2 + 26*a + 15)*29 + (3*a^8 + 4*a^7 + 28*a^6 + 5*a^5 + 23*a^4 + 6*a^3 + 13*a^2 + 3*a + 19)*29^2 + (17*a^9 + 19*a^8 + 13*a^7 + 4*a^6 + 18*a^5 + 17*a^4 + 9*a^3 + 13*a^2 + 3*a + 14)*29^3 + (26*a^9 + 3*a^8 + 22*a^7 + 11*a^6 + 8*a^4 + 15*a^3 + 16*a^2 + 23*a + 9)*29^4+O(29^5) $r_{ 6 }$ $=$ $$7 a^{9} + 18 a^{8} + 2 a^{7} + 7 a^{6} + 11 a^{5} + 23 a^{4} + 10 a^{3} + 9 a^{2} + 27 + \left(8 a^{9} + 6 a^{8} + 22 a^{7} + 19 a^{6} + 15 a^{5} + 17 a^{4} + 11 a^{3} + 10 a^{2} + 9 a + 20\right)\cdot 29 + \left(12 a^{9} + 6 a^{8} + 11 a^{7} + 14 a^{6} + 20 a^{5} + 27 a^{4} + 3 a^{3} + 28 a^{2} + 14 a + 5\right)\cdot 29^{2} + \left(22 a^{9} + 2 a^{8} + 9 a^{7} + 2 a^{6} + 26 a^{5} + 28 a^{4} + 27 a^{3} + 5 a^{2} + 19 a + 14\right)\cdot 29^{3} + \left(10 a^{9} + 26 a^{8} + 21 a^{7} + 13 a^{6} + 18 a^{5} + 18 a^{4} + 9 a^{3} + 17 a^{2} + 3 a + 23\right)\cdot 29^{4} +O(29^{5})$$ 7*a^9 + 18*a^8 + 2*a^7 + 7*a^6 + 11*a^5 + 23*a^4 + 10*a^3 + 9*a^2 + 27 + (8*a^9 + 6*a^8 + 22*a^7 + 19*a^6 + 15*a^5 + 17*a^4 + 11*a^3 + 10*a^2 + 9*a + 20)*29 + (12*a^9 + 6*a^8 + 11*a^7 + 14*a^6 + 20*a^5 + 27*a^4 + 3*a^3 + 28*a^2 + 14*a + 5)*29^2 + (22*a^9 + 2*a^8 + 9*a^7 + 2*a^6 + 26*a^5 + 28*a^4 + 27*a^3 + 5*a^2 + 19*a + 14)*29^3 + (10*a^9 + 26*a^8 + 21*a^7 + 13*a^6 + 18*a^5 + 18*a^4 + 9*a^3 + 17*a^2 + 3*a + 23)*29^4+O(29^5) $r_{ 7 }$ $=$ $$7 a^{9} + 26 a^{8} + 18 a^{7} + 7 a^{6} + 7 a^{5} + 23 a^{4} + 2 a^{3} + a^{2} + a + 3 + \left(10 a^{9} + 19 a^{8} + 10 a^{7} + 8 a^{6} + 18 a^{5} + 26 a^{4} + 2 a^{3} + a^{2} + 11 a + 18\right)\cdot 29 + \left(8 a^{9} + 3 a^{8} + 28 a^{7} + 13 a^{6} + 21 a^{5} + 12 a^{4} + 15 a^{3} + 18 a^{2} + 2 a + 23\right)\cdot 29^{2} + \left(7 a^{9} + 18 a^{8} + 17 a^{7} + 26 a^{6} + 15 a^{5} + 11 a^{3} + 18 a^{2} + 28 a + 23\right)\cdot 29^{3} + \left(a^{9} + 18 a^{8} + 10 a^{7} + 13 a^{6} + 2 a^{5} + 16 a^{4} + 2 a^{2} + 13 a + 3\right)\cdot 29^{4} +O(29^{5})$$ 7*a^9 + 26*a^8 + 18*a^7 + 7*a^6 + 7*a^5 + 23*a^4 + 2*a^3 + a^2 + a + 3 + (10*a^9 + 19*a^8 + 10*a^7 + 8*a^6 + 18*a^5 + 26*a^4 + 2*a^3 + a^2 + 11*a + 18)*29 + (8*a^9 + 3*a^8 + 28*a^7 + 13*a^6 + 21*a^5 + 12*a^4 + 15*a^3 + 18*a^2 + 2*a + 23)*29^2 + (7*a^9 + 18*a^8 + 17*a^7 + 26*a^6 + 15*a^5 + 11*a^3 + 18*a^2 + 28*a + 23)*29^3 + (a^9 + 18*a^8 + 10*a^7 + 13*a^6 + 2*a^5 + 16*a^4 + 2*a^2 + 13*a + 3)*29^4+O(29^5) $r_{ 8 }$ $=$ $$7 a^{9} + 18 a^{8} + 14 a^{7} + 14 a^{6} + 15 a^{5} + 7 a^{4} + 16 a^{3} + 20 a^{2} + 10 a + 15 + \left(3 a^{9} + 7 a^{8} + a^{7} + 12 a^{6} + 5 a^{5} + 25 a^{4} + 7 a^{3} + 21 a^{2} + 26 a + 26\right)\cdot 29 + \left(10 a^{9} + 17 a^{8} + 14 a^{7} + 28 a^{6} + 26 a^{5} + 27 a^{4} + 27 a^{3} + 26 a^{2} + 26 a + 17\right)\cdot 29^{2} + \left(16 a^{9} + 3 a^{8} + 17 a^{7} + 20 a^{6} + 15 a^{5} + a^{4} + 22 a^{3} + 17 a^{2} + 6 a + 1\right)\cdot 29^{3} + \left(3 a^{9} + 8 a^{8} + 18 a^{7} + 21 a^{6} + 18 a^{5} + 12 a^{4} + 15 a^{3} + 7 a^{2} + 10 a + 23\right)\cdot 29^{4} +O(29^{5})$$ 7*a^9 + 18*a^8 + 14*a^7 + 14*a^6 + 15*a^5 + 7*a^4 + 16*a^3 + 20*a^2 + 10*a + 15 + (3*a^9 + 7*a^8 + a^7 + 12*a^6 + 5*a^5 + 25*a^4 + 7*a^3 + 21*a^2 + 26*a + 26)*29 + (10*a^9 + 17*a^8 + 14*a^7 + 28*a^6 + 26*a^5 + 27*a^4 + 27*a^3 + 26*a^2 + 26*a + 17)*29^2 + (16*a^9 + 3*a^8 + 17*a^7 + 20*a^6 + 15*a^5 + a^4 + 22*a^3 + 17*a^2 + 6*a + 1)*29^3 + (3*a^9 + 8*a^8 + 18*a^7 + 21*a^6 + 18*a^5 + 12*a^4 + 15*a^3 + 7*a^2 + 10*a + 23)*29^4+O(29^5) $r_{ 9 }$ $=$ $$8 a^{9} + 7 a^{8} + 25 a^{7} + 2 a^{5} + 14 a^{4} + 11 a^{3} + 7 a^{2} + 22 a + 22 + \left(16 a^{9} + 8 a^{8} + 15 a^{7} + 23 a^{6} + 17 a^{5} + 3 a^{4} + 2 a^{3} + 2 a^{2} + 11 a + 18\right)\cdot 29 + \left(15 a^{9} + 9 a^{8} + 24 a^{7} + 3 a^{6} + 12 a^{5} + 18 a^{4} + 13 a^{3} + 4 a^{2} + 11 a + 20\right)\cdot 29^{2} + \left(28 a^{9} + 18 a^{8} + 21 a^{7} + 10 a^{6} + 9 a^{5} + 8 a^{4} + 10 a^{3} + 6 a^{2} + 5 a + 3\right)\cdot 29^{3} + \left(28 a^{9} + 13 a^{8} + 21 a^{7} + 26 a^{6} + 10 a^{5} + 3 a^{4} + 27 a^{3} + 23 a^{2} + 5 a + 10\right)\cdot 29^{4} +O(29^{5})$$ 8*a^9 + 7*a^8 + 25*a^7 + 2*a^5 + 14*a^4 + 11*a^3 + 7*a^2 + 22*a + 22 + (16*a^9 + 8*a^8 + 15*a^7 + 23*a^6 + 17*a^5 + 3*a^4 + 2*a^3 + 2*a^2 + 11*a + 18)*29 + (15*a^9 + 9*a^8 + 24*a^7 + 3*a^6 + 12*a^5 + 18*a^4 + 13*a^3 + 4*a^2 + 11*a + 20)*29^2 + (28*a^9 + 18*a^8 + 21*a^7 + 10*a^6 + 9*a^5 + 8*a^4 + 10*a^3 + 6*a^2 + 5*a + 3)*29^3 + (28*a^9 + 13*a^8 + 21*a^7 + 26*a^6 + 10*a^5 + 3*a^4 + 27*a^3 + 23*a^2 + 5*a + 10)*29^4+O(29^5) $r_{ 10 }$ $=$ $$11 a^{9} + 4 a^{8} + 25 a^{7} + 14 a^{6} + 18 a^{5} + 24 a^{4} + 20 a^{3} + 13 a^{2} + 2 a + 6 + \left(16 a^{9} + 28 a^{8} + 4 a^{7} + 12 a^{6} + 23 a^{5} + 25 a^{4} + 24 a^{3} + 28 a^{2} + 20 a + 27\right)\cdot 29 + \left(4 a^{9} + 21 a^{7} + 20 a^{6} + 28 a^{4} + 13 a^{3} + 18 a^{2} + 14 a\right)\cdot 29^{2} + \left(7 a^{9} + 15 a^{8} + 22 a^{7} + 28 a^{6} + a^{5} + 6 a^{4} + 6 a^{3} + 14 a^{2} + 20 a + 5\right)\cdot 29^{3} + \left(10 a^{9} + 15 a^{8} + 25 a^{7} + 13 a^{6} + 23 a^{5} + 18 a^{4} + 16 a^{3} + 10 a^{2} + 28 a + 22\right)\cdot 29^{4} +O(29^{5})$$ 11*a^9 + 4*a^8 + 25*a^7 + 14*a^6 + 18*a^5 + 24*a^4 + 20*a^3 + 13*a^2 + 2*a + 6 + (16*a^9 + 28*a^8 + 4*a^7 + 12*a^6 + 23*a^5 + 25*a^4 + 24*a^3 + 28*a^2 + 20*a + 27)*29 + (4*a^9 + 21*a^7 + 20*a^6 + 28*a^4 + 13*a^3 + 18*a^2 + 14*a)*29^2 + (7*a^9 + 15*a^8 + 22*a^7 + 28*a^6 + a^5 + 6*a^4 + 6*a^3 + 14*a^2 + 20*a + 5)*29^3 + (10*a^9 + 15*a^8 + 25*a^7 + 13*a^6 + 23*a^5 + 18*a^4 + 16*a^3 + 10*a^2 + 28*a + 22)*29^4+O(29^5) $r_{ 11 }$ $=$ $$22 a^{9} + 21 a^{8} + 10 a^{7} + 6 a^{5} + 20 a^{4} + 4 a^{3} + 5 a^{2} + 17 a + 17 + \left(24 a^{9} + 9 a^{8} + 13 a^{7} + 20 a^{6} + 25 a^{5} + 7 a^{4} + 27 a^{3} + 22 a^{2} + 15 a + 1\right)\cdot 29 + \left(22 a^{9} + 6 a^{8} + a^{6} + 28 a^{5} + 6 a^{4} + 9 a^{3} + 9 a^{2} + 7 a + 27\right)\cdot 29^{2} + \left(27 a^{9} + 28 a^{8} + 16 a^{7} + 25 a^{6} + 13 a^{5} + 9 a^{4} + a^{3} + 4 a^{2} + 16 a + 24\right)\cdot 29^{3} + \left(9 a^{9} + 10 a^{8} + 14 a^{7} + 10 a^{6} + 10 a^{5} + 19 a^{4} + 14 a^{3} + 20 a^{2} + 7 a + 16\right)\cdot 29^{4} +O(29^{5})$$ 22*a^9 + 21*a^8 + 10*a^7 + 6*a^5 + 20*a^4 + 4*a^3 + 5*a^2 + 17*a + 17 + (24*a^9 + 9*a^8 + 13*a^7 + 20*a^6 + 25*a^5 + 7*a^4 + 27*a^3 + 22*a^2 + 15*a + 1)*29 + (22*a^9 + 6*a^8 + a^6 + 28*a^5 + 6*a^4 + 9*a^3 + 9*a^2 + 7*a + 27)*29^2 + (27*a^9 + 28*a^8 + 16*a^7 + 25*a^6 + 13*a^5 + 9*a^4 + a^3 + 4*a^2 + 16*a + 24)*29^3 + (9*a^9 + 10*a^8 + 14*a^7 + 10*a^6 + 10*a^5 + 19*a^4 + 14*a^3 + 20*a^2 + 7*a + 16)*29^4+O(29^5) $r_{ 12 }$ $=$ $$2 a^{9} + 25 a^{8} + 5 a^{7} + 20 a^{6} + 26 a^{5} + 9 a^{4} + 16 a^{3} + 25 a^{2} + 6 a + 7 + \left(10 a^{9} + 5 a^{8} + 20 a^{7} + 25 a^{6} + 16 a^{5} + 3 a^{4} + 25 a^{3} + 28 a^{2} + 8 a + 19\right)\cdot 29 + \left(25 a^{9} + 13 a^{8} + 22 a^{7} + 17 a^{6} + 12 a^{5} + a^{4} + 17 a^{3} + 7 a^{2} + 22 a + 8\right)\cdot 29^{2} + \left(22 a^{9} + 4 a^{8} + 18 a^{7} + 9 a^{6} + 18 a^{5} + 27 a^{4} + 24 a^{3} + a^{2} + 2 a + 25\right)\cdot 29^{3} + \left(28 a^{9} + 8 a^{8} + 8 a^{7} + 22 a^{6} + 20 a^{5} + 21 a^{4} + 12 a^{3} + 11 a^{2} + 26 a + 2\right)\cdot 29^{4} +O(29^{5})$$ 2*a^9 + 25*a^8 + 5*a^7 + 20*a^6 + 26*a^5 + 9*a^4 + 16*a^3 + 25*a^2 + 6*a + 7 + (10*a^9 + 5*a^8 + 20*a^7 + 25*a^6 + 16*a^5 + 3*a^4 + 25*a^3 + 28*a^2 + 8*a + 19)*29 + (25*a^9 + 13*a^8 + 22*a^7 + 17*a^6 + 12*a^5 + a^4 + 17*a^3 + 7*a^2 + 22*a + 8)*29^2 + (22*a^9 + 4*a^8 + 18*a^7 + 9*a^6 + 18*a^5 + 27*a^4 + 24*a^3 + a^2 + 2*a + 25)*29^3 + (28*a^9 + 8*a^8 + 8*a^7 + 22*a^6 + 20*a^5 + 21*a^4 + 12*a^3 + 11*a^2 + 26*a + 2)*29^4+O(29^5) $r_{ 13 }$ $=$ $$24 a^{9} + 23 a^{8} + 23 a^{7} + 2 a^{6} + 16 a^{5} + 15 a^{4} + 26 a^{3} + 18 a^{2} + 23 a + 20 + \left(24 a^{9} + 26 a^{8} + 17 a^{7} + 10 a^{6} + 8 a^{5} + 28 a^{4} + 6 a^{3} + 9 a^{2} + a + 18\right)\cdot 29 + \left(15 a^{9} + 11 a^{8} + 9 a^{6} + 18 a^{5} + 21 a^{4} + 8 a^{3} + 24 a^{2} + 11 a + 14\right)\cdot 29^{2} + \left(16 a^{9} + 3 a^{8} + 12 a^{7} + 14 a^{6} + 11 a^{5} + 28 a^{4} + 26 a^{3} + 17 a^{2} + 27 a + 26\right)\cdot 29^{3} + \left(13 a^{9} + 3 a^{8} + 16 a^{7} + 6 a^{6} + 22 a^{5} + 3 a^{4} + 4 a^{3} + 28 a^{2} + 15 a + 19\right)\cdot 29^{4} +O(29^{5})$$ 24*a^9 + 23*a^8 + 23*a^7 + 2*a^6 + 16*a^5 + 15*a^4 + 26*a^3 + 18*a^2 + 23*a + 20 + (24*a^9 + 26*a^8 + 17*a^7 + 10*a^6 + 8*a^5 + 28*a^4 + 6*a^3 + 9*a^2 + a + 18)*29 + (15*a^9 + 11*a^8 + 9*a^6 + 18*a^5 + 21*a^4 + 8*a^3 + 24*a^2 + 11*a + 14)*29^2 + (16*a^9 + 3*a^8 + 12*a^7 + 14*a^6 + 11*a^5 + 28*a^4 + 26*a^3 + 17*a^2 + 27*a + 26)*29^3 + (13*a^9 + 3*a^8 + 16*a^7 + 6*a^6 + 22*a^5 + 3*a^4 + 4*a^3 + 28*a^2 + 15*a + 19)*29^4+O(29^5) $r_{ 14 }$ $=$ $$19 a^{9} + 2 a^{8} + 7 a^{7} + 5 a^{6} + 21 a^{5} + 10 a^{4} + 3 a^{3} + 27 a^{2} + 10 a + 3 + \left(14 a^{9} + 15 a^{8} + 5 a^{7} + 18 a^{6} + 16 a^{5} + 6 a^{4} + 10 a^{3} + 27 a^{2} + 17 a + 9\right)\cdot 29 + \left(21 a^{9} + 25 a^{8} + 6 a^{7} + 28 a^{6} + a^{4} + 26 a^{3} + 19 a^{2} + 24 a + 14\right)\cdot 29^{2} + \left(18 a^{9} + 27 a^{8} + 21 a^{7} + 25 a^{6} + 14 a^{5} + 25 a^{4} + 26 a^{3} + 15 a^{2} + 5 a + 17\right)\cdot 29^{3} + \left(9 a^{9} + 25 a^{8} + 24 a^{7} + 10 a^{6} + 19 a^{5} + a^{4} + 15 a^{3} + 5 a^{2} + a + 27\right)\cdot 29^{4} +O(29^{5})$$ 19*a^9 + 2*a^8 + 7*a^7 + 5*a^6 + 21*a^5 + 10*a^4 + 3*a^3 + 27*a^2 + 10*a + 3 + (14*a^9 + 15*a^8 + 5*a^7 + 18*a^6 + 16*a^5 + 6*a^4 + 10*a^3 + 27*a^2 + 17*a + 9)*29 + (21*a^9 + 25*a^8 + 6*a^7 + 28*a^6 + a^4 + 26*a^3 + 19*a^2 + 24*a + 14)*29^2 + (18*a^9 + 27*a^8 + 21*a^7 + 25*a^6 + 14*a^5 + 25*a^4 + 26*a^3 + 15*a^2 + 5*a + 17)*29^3 + (9*a^9 + 25*a^8 + 24*a^7 + 10*a^6 + 19*a^5 + a^4 + 15*a^3 + 5*a^2 + a + 27)*29^4+O(29^5) $r_{ 15 }$ $=$ $$16 a^{9} + 21 a^{8} + 3 a^{7} + 22 a^{6} + 23 a^{5} + 19 a^{4} + 18 a^{3} + 24 a^{2} + 26 a + 24 + \left(24 a^{9} + 5 a^{8} + 5 a^{7} + 4 a^{6} + a^{5} + 25 a^{4} + 14 a^{3} + 11 a^{2} + 17 a + 27\right)\cdot 29 + \left(17 a^{9} + 26 a^{8} + 4 a^{7} + 9 a^{6} + 9 a^{5} + 27 a^{4} + 11 a^{3} + 23 a^{2} + 24 a + 9\right)\cdot 29^{2} + \left(21 a^{9} + 12 a^{8} + 28 a^{7} + 11 a^{6} + a^{5} + 13 a^{4} + 18 a^{3} + a^{2} + 24 a + 20\right)\cdot 29^{3} + \left(13 a^{9} + 2 a^{8} + 14 a^{7} + 2 a^{6} + 2 a^{5} + 11 a^{4} + 25 a^{3} + 20 a^{2} + 25 a + 8\right)\cdot 29^{4} +O(29^{5})$$ 16*a^9 + 21*a^8 + 3*a^7 + 22*a^6 + 23*a^5 + 19*a^4 + 18*a^3 + 24*a^2 + 26*a + 24 + (24*a^9 + 5*a^8 + 5*a^7 + 4*a^6 + a^5 + 25*a^4 + 14*a^3 + 11*a^2 + 17*a + 27)*29 + (17*a^9 + 26*a^8 + 4*a^7 + 9*a^6 + 9*a^5 + 27*a^4 + 11*a^3 + 23*a^2 + 24*a + 9)*29^2 + (21*a^9 + 12*a^8 + 28*a^7 + 11*a^6 + a^5 + 13*a^4 + 18*a^3 + a^2 + 24*a + 20)*29^3 + (13*a^9 + 2*a^8 + 14*a^7 + 2*a^6 + 2*a^5 + 11*a^4 + 25*a^3 + 20*a^2 + 25*a + 8)*29^4+O(29^5) $r_{ 16 }$ $=$ $$3 a^{9} + 20 a^{8} + 11 a^{7} + 28 a^{6} + 28 a^{5} + 9 a^{4} + 6 a^{3} + 18 a^{2} + 20 a + 20 + \left(2 a^{9} + 24 a^{8} + a^{7} + 13 a^{6} + 15 a^{5} + 6 a^{4} + 21 a^{3} + 9 a^{2} + 24 a + 13\right)\cdot 29 + \left(17 a^{9} + 10 a^{8} + 21 a^{7} + 13 a^{6} + 2 a^{5} + 25 a^{4} + 24 a^{3} + 23 a^{2} + 19 a + 22\right)\cdot 29^{2} + \left(a^{9} + 4 a^{8} + 19 a^{7} + 21 a^{6} + 5 a^{5} + 15 a^{4} + 10 a^{3} + 14 a^{2} + 8 a + 28\right)\cdot 29^{3} + \left(17 a^{9} + 26 a^{8} + 13 a^{7} + 28 a^{6} + 25 a^{5} + 18 a^{4} + 10 a^{3} + 17 a^{2} + a + 4\right)\cdot 29^{4} +O(29^{5})$$ 3*a^9 + 20*a^8 + 11*a^7 + 28*a^6 + 28*a^5 + 9*a^4 + 6*a^3 + 18*a^2 + 20*a + 20 + (2*a^9 + 24*a^8 + a^7 + 13*a^6 + 15*a^5 + 6*a^4 + 21*a^3 + 9*a^2 + 24*a + 13)*29 + (17*a^9 + 10*a^8 + 21*a^7 + 13*a^6 + 2*a^5 + 25*a^4 + 24*a^3 + 23*a^2 + 19*a + 22)*29^2 + (a^9 + 4*a^8 + 19*a^7 + 21*a^6 + 5*a^5 + 15*a^4 + 10*a^3 + 14*a^2 + 8*a + 28)*29^3 + (17*a^9 + 26*a^8 + 13*a^7 + 28*a^6 + 25*a^5 + 18*a^4 + 10*a^3 + 17*a^2 + a + 4)*29^4+O(29^5) $r_{ 17 }$ $=$ $$25 a^{9} + 21 a^{8} + 3 a^{7} + 10 a^{6} + 20 a^{5} + 3 a^{4} + 3 a^{3} + 25 a^{2} + 6 a + 20 + \left(2 a^{9} + 28 a^{8} + 5 a^{7} + 11 a^{6} + 13 a^{5} + 19 a^{4} + 27 a^{3} + 17 a^{2} + 16 a + 1\right)\cdot 29 + \left(14 a^{9} + 8 a^{8} + 9 a^{7} + a^{6} + 3 a^{5} + 25 a^{4} + 15 a^{3} + 14 a^{2} + 28 a + 3\right)\cdot 29^{2} + \left(27 a^{9} + 14 a^{8} + 12 a^{7} + 6 a^{6} + 16 a^{5} + 28 a^{4} + 5 a^{3} + 5 a^{2} + 28 a + 27\right)\cdot 29^{3} + \left(a^{9} + 4 a^{8} + 4 a^{7} + 16 a^{6} + 12 a^{5} + 5 a^{4} + 8 a^{3} + 11 a^{2} + 21 a + 28\right)\cdot 29^{4} +O(29^{5})$$ 25*a^9 + 21*a^8 + 3*a^7 + 10*a^6 + 20*a^5 + 3*a^4 + 3*a^3 + 25*a^2 + 6*a + 20 + (2*a^9 + 28*a^8 + 5*a^7 + 11*a^6 + 13*a^5 + 19*a^4 + 27*a^3 + 17*a^2 + 16*a + 1)*29 + (14*a^9 + 8*a^8 + 9*a^7 + a^6 + 3*a^5 + 25*a^4 + 15*a^3 + 14*a^2 + 28*a + 3)*29^2 + (27*a^9 + 14*a^8 + 12*a^7 + 6*a^6 + 16*a^5 + 28*a^4 + 5*a^3 + 5*a^2 + 28*a + 27)*29^3 + (a^9 + 4*a^8 + 4*a^7 + 16*a^6 + 12*a^5 + 5*a^4 + 8*a^3 + 11*a^2 + 21*a + 28)*29^4+O(29^5) $r_{ 18 }$ $=$ $$26 a^{9} + 6 a^{8} + 10 a^{7} + 11 a^{6} + 26 a^{4} + 24 a^{3} + 6 a^{2} + 19 a + 9 + \left(20 a^{9} + 2 a^{8} + 23 a^{7} + 9 a^{6} + 8 a^{5} + 12 a^{4} + a^{3} + 11 a^{2} + 15 a + 23\right)\cdot 29 + \left(22 a^{9} + 16 a^{8} + 2 a^{7} + 2 a^{6} + 2 a^{5} + 5 a^{4} + 23 a^{3} + 21 a^{2} + 19 a + 12\right)\cdot 29^{2} + \left(26 a^{9} + 22 a^{8} + 24 a^{7} + 16 a^{6} + 11 a^{5} + 14 a^{4} + 22 a^{3} + 16 a^{2} + 4 a + 10\right)\cdot 29^{3} + \left(4 a^{9} + 4 a^{8} + 3 a^{6} + 2 a^{5} + 27 a^{4} + 14 a^{3} + 20 a^{2} + 12 a + 25\right)\cdot 29^{4} +O(29^{5})$$ 26*a^9 + 6*a^8 + 10*a^7 + 11*a^6 + 26*a^4 + 24*a^3 + 6*a^2 + 19*a + 9 + (20*a^9 + 2*a^8 + 23*a^7 + 9*a^6 + 8*a^5 + 12*a^4 + a^3 + 11*a^2 + 15*a + 23)*29 + (22*a^9 + 16*a^8 + 2*a^7 + 2*a^6 + 2*a^5 + 5*a^4 + 23*a^3 + 21*a^2 + 19*a + 12)*29^2 + (26*a^9 + 22*a^8 + 24*a^7 + 16*a^6 + 11*a^5 + 14*a^4 + 22*a^3 + 16*a^2 + 4*a + 10)*29^3 + (4*a^9 + 4*a^8 + 3*a^6 + 2*a^5 + 27*a^4 + 14*a^3 + 20*a^2 + 12*a + 25)*29^4+O(29^5) $r_{ 19 }$ $=$ $$26 a^{8} + 5 a^{7} + 23 a^{6} + 22 a^{5} + 17 a^{4} + 28 a^{3} + 28 a^{2} + 26 a + 16 + \left(28 a^{9} + 24 a^{8} + 17 a^{7} + a^{6} + 2 a^{5} + 26 a^{4} + 17 a^{3} + 18 a^{2} + 10 a + 3\right)\cdot 29 + \left(7 a^{9} + 9 a^{8} + 23 a^{7} + a^{5} + 22 a^{4} + 20 a^{3} + 16 a^{2} + 15 a + 7\right)\cdot 29^{2} + \left(15 a^{9} + 8 a^{8} + 26 a^{7} + 18 a^{6} + 22 a^{5} + 4 a^{4} + 24 a^{3} + 26 a^{2} + 3 a + 25\right)\cdot 29^{3} + \left(16 a^{9} + 19 a^{8} + 11 a^{7} + 20 a^{6} + 6 a^{4} + 18 a^{3} + 8 a^{2} + 12 a + 13\right)\cdot 29^{4} +O(29^{5})$$ 26*a^8 + 5*a^7 + 23*a^6 + 22*a^5 + 17*a^4 + 28*a^3 + 28*a^2 + 26*a + 16 + (28*a^9 + 24*a^8 + 17*a^7 + a^6 + 2*a^5 + 26*a^4 + 17*a^3 + 18*a^2 + 10*a + 3)*29 + (7*a^9 + 9*a^8 + 23*a^7 + a^5 + 22*a^4 + 20*a^3 + 16*a^2 + 15*a + 7)*29^2 + (15*a^9 + 8*a^8 + 26*a^7 + 18*a^6 + 22*a^5 + 4*a^4 + 24*a^3 + 26*a^2 + 3*a + 25)*29^3 + (16*a^9 + 19*a^8 + 11*a^7 + 20*a^6 + 6*a^4 + 18*a^3 + 8*a^2 + 12*a + 13)*29^4+O(29^5) $r_{ 20 }$ $=$ $$13 a^{9} + 17 a^{8} + 21 a^{7} + 4 a^{6} + 27 a^{5} + 7 a^{4} + 14 a^{3} + 7 a^{2} + 7 a + 6 + \left(5 a^{9} + 15 a^{8} + 14 a^{7} + 23 a^{6} + 12 a^{5} + 14 a^{4} + 3 a^{3} + 20 a^{2} + 13 a + 5\right)\cdot 29 + \left(a^{8} + 10 a^{7} + 19 a^{6} + 12 a^{5} + 3 a^{4} + 24 a^{3} + 5 a^{2} + 8 a + 7\right)\cdot 29^{2} + \left(a^{9} + 5 a^{8} + 23 a^{7} + 2 a^{6} + 9 a^{5} + 18 a^{4} + 18 a^{3} + 3 a^{2} + 14 a + 16\right)\cdot 29^{3} + \left(27 a^{9} + 11 a^{8} + 25 a^{7} + 17 a^{6} + 17 a^{5} + 24 a^{4} + 28 a^{3} + 16 a^{2} + 24 a + 16\right)\cdot 29^{4} +O(29^{5})$$ 13*a^9 + 17*a^8 + 21*a^7 + 4*a^6 + 27*a^5 + 7*a^4 + 14*a^3 + 7*a^2 + 7*a + 6 + (5*a^9 + 15*a^8 + 14*a^7 + 23*a^6 + 12*a^5 + 14*a^4 + 3*a^3 + 20*a^2 + 13*a + 5)*29 + (a^8 + 10*a^7 + 19*a^6 + 12*a^5 + 3*a^4 + 24*a^3 + 5*a^2 + 8*a + 7)*29^2 + (a^9 + 5*a^8 + 23*a^7 + 2*a^6 + 9*a^5 + 18*a^4 + 18*a^3 + 3*a^2 + 14*a + 16)*29^3 + (27*a^9 + 11*a^8 + 25*a^7 + 17*a^6 + 17*a^5 + 24*a^4 + 28*a^3 + 16*a^2 + 24*a + 16)*29^4+O(29^5)

Generators of the action on the roots $r_1, \ldots, r_{ 20 }$

 Cycle notation $(1,4,18,15)(2,3,17,16)(5,12,14,7)(6,11,13,8)(9,20,10,19)$ $(1,7,10,8,17,15,14,20,13,3,18,12,9,11,2,4,5,19,6,16)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 20 }$ Character values $c1$ $c2$ $c3$ $c4$ $c5$ $c6$ $c7$ $c8$ $1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,20)$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $1$ $4$ $(1,4,18,15)(2,3,17,16)(5,12,14,7)(6,11,13,8)(9,20,10,19)$ $\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $1$ $4$ $(1,15,18,4)(2,16,17,3)(5,7,14,12)(6,8,13,11)(9,19,10,20)$ $-\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $-\zeta_{20}^{5}$ $\zeta_{20}^{5}$ $1$ $5$ $(1,17,13,9,5)(2,6,10,14,18)(3,11,19,7,15)(4,16,8,20,12)$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $-\zeta_{20}^{6}$ $\zeta_{20}^{4}$ $-\zeta_{20}^{2}$ $-\zeta_{20}^{2}$ $\zeta_{20}^{4}$ $-\zeta_{20}^{6}$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $1$ $5$ $(1,13,5,17,9)(2,10,18,6,14)(3,19,15,11,7)(4,8,12,16,20)$ $-\zeta_{20}^{6}$ $-\zeta_{20}^{2}$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $\zeta_{20}^{4}$ $\zeta_{20}^{4}$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $-\zeta_{20}^{2}$ $-\zeta_{20}^{6}$ $1$ $5$ $(1,9,17,5,13)(2,14,6,18,10)(3,7,11,15,19)(4,20,16,12,8)$ $\zeta_{20}^{4}$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $-\zeta_{20}^{2}$ $-\zeta_{20}^{6}$ $-\zeta_{20}^{6}$ $-\zeta_{20}^{2}$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $\zeta_{20}^{4}$ $1$ $5$ $(1,5,9,13,17)(2,18,14,10,6)(3,15,7,19,11)(4,12,20,8,16)$ $-\zeta_{20}^{2}$ $\zeta_{20}^{4}$ $-\zeta_{20}^{6}$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$ $-\zeta_{20}^{6}$ $\zeta_{20}^{4}$ $-\zeta_{20}^{2}$ $1$ $10$ $(1,10,17,14,13,18,9,2,5,6)(3,12,11,4,19,16,7,8,15,20)$ $-\zeta_{20}^{4}$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $\zeta_{20}^{2}$ $\zeta_{20}^{6}$ $\zeta_{20}^{6}$ $\zeta_{20}^{2}$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $-\zeta_{20}^{4}$ $1$ $10$ $(1,14,9,6,17,18,5,10,13,2)(3,4,7,20,11,16,15,12,19,8)$ $\zeta_{20}^{2}$ $-\zeta_{20}^{4}$ $\zeta_{20}^{6}$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $\zeta_{20}^{6}$ $-\zeta_{20}^{4}$ $\zeta_{20}^{2}$ $1$ $10$ $(1,2,13,10,5,18,17,6,9,14)(3,8,19,12,15,16,11,20,7,4)$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $\zeta_{20}^{6}$ $-\zeta_{20}^{4}$ $\zeta_{20}^{2}$ $\zeta_{20}^{2}$ $-\zeta_{20}^{4}$ $\zeta_{20}^{6}$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $1$ $10$ $(1,6,5,2,9,18,13,14,17,10)(3,20,15,8,7,16,19,4,11,12)$ $\zeta_{20}^{6}$ $\zeta_{20}^{2}$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $-\zeta_{20}^{4}$ $-\zeta_{20}^{4}$ $-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$ $\zeta_{20}^{2}$ $\zeta_{20}^{6}$ $1$ $20$ $(1,7,10,8,17,15,14,20,13,3,18,12,9,11,2,4,5,19,6,16)$ $\zeta_{20}^{7}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $\zeta_{20}$ $\zeta_{20}^{3}$ $-\zeta_{20}^{3}$ $-\zeta_{20}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $-\zeta_{20}^{7}$ $1$ $20$ $(1,8,14,3,9,4,6,7,17,20,18,11,5,16,10,15,13,12,2,19)$ $\zeta_{20}$ $\zeta_{20}^{7}$ $\zeta_{20}^{3}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $-\zeta_{20}^{3}$ $-\zeta_{20}^{7}$ $-\zeta_{20}$ $1$ $20$ $(1,20,2,7,13,4,10,3,5,8,18,19,17,12,6,15,9,16,14,11)$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $\zeta_{20}^{3}$ $\zeta_{20}^{7}$ $\zeta_{20}$ $-\zeta_{20}$ $-\zeta_{20}^{7}$ $-\zeta_{20}^{3}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $1$ $20$ $(1,3,6,20,5,15,2,8,9,7,18,16,13,19,14,4,17,11,10,12)$ $\zeta_{20}^{3}$ $\zeta_{20}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $\zeta_{20}^{7}$ $-\zeta_{20}^{7}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $-\zeta_{20}$ $-\zeta_{20}^{3}$ $1$ $20$ $(1,12,10,11,17,4,14,19,13,16,18,7,9,8,2,15,5,20,6,3)$ $-\zeta_{20}^{7}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $-\zeta_{20}$ $-\zeta_{20}^{3}$ $\zeta_{20}^{3}$ $\zeta_{20}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $\zeta_{20}^{7}$ $1$ $20$ $(1,11,14,16,9,15,6,12,17,19,18,8,5,3,10,4,13,7,2,20)$ $-\zeta_{20}$ $-\zeta_{20}^{7}$ $-\zeta_{20}^{3}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $\zeta_{20}^{3}$ $\zeta_{20}^{7}$ $\zeta_{20}$ $1$ $20$ $(1,19,2,12,13,15,10,16,5,11,18,20,17,7,6,4,9,3,14,8)$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $-\zeta_{20}^{3}$ $-\zeta_{20}^{7}$ $-\zeta_{20}$ $\zeta_{20}$ $\zeta_{20}^{7}$ $\zeta_{20}^{3}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $1$ $20$ $(1,16,6,19,5,4,2,11,9,12,18,3,13,20,14,15,17,8,10,7)$ $-\zeta_{20}^{3}$ $-\zeta_{20}$ $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ $-\zeta_{20}^{7}$ $\zeta_{20}^{7}$ $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ $\zeta_{20}$ $\zeta_{20}^{3}$
The blue line marks the conjugacy class containing complex conjugation.