Properties

Label 1.25.20t1.a.e
Dimension $1$
Group $C_{20}$
Conductor $25$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: 20T1
Conductor: \(25\)\(\medspace = 5^{2} \)
Artin field: Galois closure of \(\Q(\zeta_{25})\)
Galois orbit size: $8$
Smallest permutation container: 20T1
Parity: odd
Dirichlet character: \(\chi_{25}(13,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{10} + x^{6} + 25x^{5} + 8x^{4} + 17x^{3} + 2x^{2} + 22x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 14 a^{9} + 24 a^{8} + 21 a^{7} + 19 a^{6} + 26 a^{5} + 18 a^{3} + 20 a^{2} + 7 a + 22 + \left(21 a^{9} + 6 a^{8} + 12 a^{7} + 24 a^{6} + 15 a^{5} + 14 a^{4} + 17 a^{3} + 24 a^{2} + 2 a + 3\right)\cdot 29 + \left(11 a^{9} + 24 a^{8} + 19 a^{7} + 14 a^{6} + 18 a^{5} + 26 a^{4} + 14 a^{3} + a^{2} + 3 a\right)\cdot 29^{2} + \left(26 a^{9} + 2 a^{8} + 27 a^{7} + 22 a^{6} + 2 a^{5} + 2 a^{4} + 6 a^{3} + 15 a^{2} + 22 a + 15\right)\cdot 29^{3} + \left(15 a^{9} + 4 a^{8} + 21 a^{7} + 26 a^{6} + 12 a^{5} + 7 a^{4} + 2 a^{3} + 7 a^{2} + 20 a + 18\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a^{9} + 28 a^{8} + 14 a^{7} + 21 a^{6} + 8 a^{5} + 4 a^{4} + a^{3} + 3 a^{2} + 27 a + 13 + \left(26 a^{9} + 5 a^{8} + 2 a^{7} + 27 a^{6} + 23 a^{5} + 24 a^{4} + 10 a^{3} + 9 a^{2} + 24 a + 6\right)\cdot 29 + \left(25 a^{9} + 9 a^{8} + 16 a^{7} + 20 a^{6} + 4 a^{5} + 23 a^{4} + 20 a^{3} + 27 a^{2} + 13 a + 24\right)\cdot 29^{2} + \left(11 a^{9} + 19 a^{8} + 9 a^{7} + 13 a^{6} + 5 a^{5} + 11 a^{4} + 3 a^{3} + 4 a^{2} + 7 a + 10\right)\cdot 29^{3} + \left(22 a^{9} + 14 a^{8} + 14 a^{7} + 16 a^{6} + 23 a^{5} + 20 a^{4} + a^{3} + 4 a^{2} + 12 a + 17\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 a^{9} + 22 a^{8} + 22 a^{7} + 6 a^{6} + 8 a^{4} + 6 a^{3} + 17 a + 27 + \left(28 a^{9} + 26 a^{8} + 11 a^{7} + 23 a^{6} + 10 a^{5} + 25 a^{3} + 4 a^{2} + 2 a + 6\right)\cdot 29 + \left(11 a^{8} + a^{7} + 4 a^{6} + 26 a^{5} + 17 a^{4} + 19 a^{2} + 8 a + 19\right)\cdot 29^{2} + \left(15 a^{9} + 19 a^{8} + 27 a^{7} + 6 a^{6} + 4 a^{5} + 2 a^{3} + 6 a^{2} + 14 a + 21\right)\cdot 29^{3} + \left(16 a^{9} + 6 a^{8} + 5 a^{7} + 10 a^{6} + 13 a^{5} + 5 a^{4} + 28 a^{3} + 6 a^{2} + 27 a + 14\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a^{9} + 7 a^{8} + 7 a^{7} + 21 a^{6} + 20 a^{5} + 26 a^{4} + 6 a^{3} + 17 a^{2} + 15 a + 10 + \left(8 a^{9} + 4 a^{8} + 20 a^{7} + 11 a^{6} + 6 a^{5} + 8 a^{4} + 6 a^{2} + 14 a + 22\right)\cdot 29 + \left(5 a^{9} + 15 a^{8} + 18 a^{7} + 7 a^{6} + 4 a^{5} + 22 a^{3} + 23 a^{2} + 9 a + 1\right)\cdot 29^{2} + \left(16 a^{9} + 11 a^{8} + 7 a^{7} + 3 a^{6} + 9 a^{5} + 24 a^{4} + 9 a^{3} + 20 a^{2} + 25 a + 15\right)\cdot 29^{3} + \left(10 a^{9} + 4 a^{8} + 20 a^{7} + 26 a^{6} + 5 a^{5} + 9 a^{4} + 19 a^{3} + 5 a^{2} + 24 a + 10\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a^{9} + 12 a^{8} + 15 a^{7} + 27 a^{6} + 23 a^{5} + 26 a^{4} + 17 a^{2} + 3 + \left(21 a^{9} + 16 a^{8} + 6 a^{7} + 17 a^{6} + 2 a^{5} + 21 a^{4} + 4 a^{3} + 3 a^{2} + 26 a + 15\right)\cdot 29 + \left(3 a^{8} + 4 a^{7} + 28 a^{6} + 5 a^{5} + 23 a^{4} + 6 a^{3} + 13 a^{2} + 3 a + 19\right)\cdot 29^{2} + \left(17 a^{9} + 19 a^{8} + 13 a^{7} + 4 a^{6} + 18 a^{5} + 17 a^{4} + 9 a^{3} + 13 a^{2} + 3 a + 14\right)\cdot 29^{3} + \left(26 a^{9} + 3 a^{8} + 22 a^{7} + 11 a^{6} + 8 a^{4} + 15 a^{3} + 16 a^{2} + 23 a + 9\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a^{9} + 18 a^{8} + 2 a^{7} + 7 a^{6} + 11 a^{5} + 23 a^{4} + 10 a^{3} + 9 a^{2} + 27 + \left(8 a^{9} + 6 a^{8} + 22 a^{7} + 19 a^{6} + 15 a^{5} + 17 a^{4} + 11 a^{3} + 10 a^{2} + 9 a + 20\right)\cdot 29 + \left(12 a^{9} + 6 a^{8} + 11 a^{7} + 14 a^{6} + 20 a^{5} + 27 a^{4} + 3 a^{3} + 28 a^{2} + 14 a + 5\right)\cdot 29^{2} + \left(22 a^{9} + 2 a^{8} + 9 a^{7} + 2 a^{6} + 26 a^{5} + 28 a^{4} + 27 a^{3} + 5 a^{2} + 19 a + 14\right)\cdot 29^{3} + \left(10 a^{9} + 26 a^{8} + 21 a^{7} + 13 a^{6} + 18 a^{5} + 18 a^{4} + 9 a^{3} + 17 a^{2} + 3 a + 23\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 7 a^{9} + 26 a^{8} + 18 a^{7} + 7 a^{6} + 7 a^{5} + 23 a^{4} + 2 a^{3} + a^{2} + a + 3 + \left(10 a^{9} + 19 a^{8} + 10 a^{7} + 8 a^{6} + 18 a^{5} + 26 a^{4} + 2 a^{3} + a^{2} + 11 a + 18\right)\cdot 29 + \left(8 a^{9} + 3 a^{8} + 28 a^{7} + 13 a^{6} + 21 a^{5} + 12 a^{4} + 15 a^{3} + 18 a^{2} + 2 a + 23\right)\cdot 29^{2} + \left(7 a^{9} + 18 a^{8} + 17 a^{7} + 26 a^{6} + 15 a^{5} + 11 a^{3} + 18 a^{2} + 28 a + 23\right)\cdot 29^{3} + \left(a^{9} + 18 a^{8} + 10 a^{7} + 13 a^{6} + 2 a^{5} + 16 a^{4} + 2 a^{2} + 13 a + 3\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 7 a^{9} + 18 a^{8} + 14 a^{7} + 14 a^{6} + 15 a^{5} + 7 a^{4} + 16 a^{3} + 20 a^{2} + 10 a + 15 + \left(3 a^{9} + 7 a^{8} + a^{7} + 12 a^{6} + 5 a^{5} + 25 a^{4} + 7 a^{3} + 21 a^{2} + 26 a + 26\right)\cdot 29 + \left(10 a^{9} + 17 a^{8} + 14 a^{7} + 28 a^{6} + 26 a^{5} + 27 a^{4} + 27 a^{3} + 26 a^{2} + 26 a + 17\right)\cdot 29^{2} + \left(16 a^{9} + 3 a^{8} + 17 a^{7} + 20 a^{6} + 15 a^{5} + a^{4} + 22 a^{3} + 17 a^{2} + 6 a + 1\right)\cdot 29^{3} + \left(3 a^{9} + 8 a^{8} + 18 a^{7} + 21 a^{6} + 18 a^{5} + 12 a^{4} + 15 a^{3} + 7 a^{2} + 10 a + 23\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 8 a^{9} + 7 a^{8} + 25 a^{7} + 2 a^{5} + 14 a^{4} + 11 a^{3} + 7 a^{2} + 22 a + 22 + \left(16 a^{9} + 8 a^{8} + 15 a^{7} + 23 a^{6} + 17 a^{5} + 3 a^{4} + 2 a^{3} + 2 a^{2} + 11 a + 18\right)\cdot 29 + \left(15 a^{9} + 9 a^{8} + 24 a^{7} + 3 a^{6} + 12 a^{5} + 18 a^{4} + 13 a^{3} + 4 a^{2} + 11 a + 20\right)\cdot 29^{2} + \left(28 a^{9} + 18 a^{8} + 21 a^{7} + 10 a^{6} + 9 a^{5} + 8 a^{4} + 10 a^{3} + 6 a^{2} + 5 a + 3\right)\cdot 29^{3} + \left(28 a^{9} + 13 a^{8} + 21 a^{7} + 26 a^{6} + 10 a^{5} + 3 a^{4} + 27 a^{3} + 23 a^{2} + 5 a + 10\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 11 a^{9} + 4 a^{8} + 25 a^{7} + 14 a^{6} + 18 a^{5} + 24 a^{4} + 20 a^{3} + 13 a^{2} + 2 a + 6 + \left(16 a^{9} + 28 a^{8} + 4 a^{7} + 12 a^{6} + 23 a^{5} + 25 a^{4} + 24 a^{3} + 28 a^{2} + 20 a + 27\right)\cdot 29 + \left(4 a^{9} + 21 a^{7} + 20 a^{6} + 28 a^{4} + 13 a^{3} + 18 a^{2} + 14 a\right)\cdot 29^{2} + \left(7 a^{9} + 15 a^{8} + 22 a^{7} + 28 a^{6} + a^{5} + 6 a^{4} + 6 a^{3} + 14 a^{2} + 20 a + 5\right)\cdot 29^{3} + \left(10 a^{9} + 15 a^{8} + 25 a^{7} + 13 a^{6} + 23 a^{5} + 18 a^{4} + 16 a^{3} + 10 a^{2} + 28 a + 22\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 22 a^{9} + 21 a^{8} + 10 a^{7} + 6 a^{5} + 20 a^{4} + 4 a^{3} + 5 a^{2} + 17 a + 17 + \left(24 a^{9} + 9 a^{8} + 13 a^{7} + 20 a^{6} + 25 a^{5} + 7 a^{4} + 27 a^{3} + 22 a^{2} + 15 a + 1\right)\cdot 29 + \left(22 a^{9} + 6 a^{8} + a^{6} + 28 a^{5} + 6 a^{4} + 9 a^{3} + 9 a^{2} + 7 a + 27\right)\cdot 29^{2} + \left(27 a^{9} + 28 a^{8} + 16 a^{7} + 25 a^{6} + 13 a^{5} + 9 a^{4} + a^{3} + 4 a^{2} + 16 a + 24\right)\cdot 29^{3} + \left(9 a^{9} + 10 a^{8} + 14 a^{7} + 10 a^{6} + 10 a^{5} + 19 a^{4} + 14 a^{3} + 20 a^{2} + 7 a + 16\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 2 a^{9} + 25 a^{8} + 5 a^{7} + 20 a^{6} + 26 a^{5} + 9 a^{4} + 16 a^{3} + 25 a^{2} + 6 a + 7 + \left(10 a^{9} + 5 a^{8} + 20 a^{7} + 25 a^{6} + 16 a^{5} + 3 a^{4} + 25 a^{3} + 28 a^{2} + 8 a + 19\right)\cdot 29 + \left(25 a^{9} + 13 a^{8} + 22 a^{7} + 17 a^{6} + 12 a^{5} + a^{4} + 17 a^{3} + 7 a^{2} + 22 a + 8\right)\cdot 29^{2} + \left(22 a^{9} + 4 a^{8} + 18 a^{7} + 9 a^{6} + 18 a^{5} + 27 a^{4} + 24 a^{3} + a^{2} + 2 a + 25\right)\cdot 29^{3} + \left(28 a^{9} + 8 a^{8} + 8 a^{7} + 22 a^{6} + 20 a^{5} + 21 a^{4} + 12 a^{3} + 11 a^{2} + 26 a + 2\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 13 }$ $=$ \( 24 a^{9} + 23 a^{8} + 23 a^{7} + 2 a^{6} + 16 a^{5} + 15 a^{4} + 26 a^{3} + 18 a^{2} + 23 a + 20 + \left(24 a^{9} + 26 a^{8} + 17 a^{7} + 10 a^{6} + 8 a^{5} + 28 a^{4} + 6 a^{3} + 9 a^{2} + a + 18\right)\cdot 29 + \left(15 a^{9} + 11 a^{8} + 9 a^{6} + 18 a^{5} + 21 a^{4} + 8 a^{3} + 24 a^{2} + 11 a + 14\right)\cdot 29^{2} + \left(16 a^{9} + 3 a^{8} + 12 a^{7} + 14 a^{6} + 11 a^{5} + 28 a^{4} + 26 a^{3} + 17 a^{2} + 27 a + 26\right)\cdot 29^{3} + \left(13 a^{9} + 3 a^{8} + 16 a^{7} + 6 a^{6} + 22 a^{5} + 3 a^{4} + 4 a^{3} + 28 a^{2} + 15 a + 19\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 14 }$ $=$ \( 19 a^{9} + 2 a^{8} + 7 a^{7} + 5 a^{6} + 21 a^{5} + 10 a^{4} + 3 a^{3} + 27 a^{2} + 10 a + 3 + \left(14 a^{9} + 15 a^{8} + 5 a^{7} + 18 a^{6} + 16 a^{5} + 6 a^{4} + 10 a^{3} + 27 a^{2} + 17 a + 9\right)\cdot 29 + \left(21 a^{9} + 25 a^{8} + 6 a^{7} + 28 a^{6} + a^{4} + 26 a^{3} + 19 a^{2} + 24 a + 14\right)\cdot 29^{2} + \left(18 a^{9} + 27 a^{8} + 21 a^{7} + 25 a^{6} + 14 a^{5} + 25 a^{4} + 26 a^{3} + 15 a^{2} + 5 a + 17\right)\cdot 29^{3} + \left(9 a^{9} + 25 a^{8} + 24 a^{7} + 10 a^{6} + 19 a^{5} + a^{4} + 15 a^{3} + 5 a^{2} + a + 27\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 15 }$ $=$ \( 16 a^{9} + 21 a^{8} + 3 a^{7} + 22 a^{6} + 23 a^{5} + 19 a^{4} + 18 a^{3} + 24 a^{2} + 26 a + 24 + \left(24 a^{9} + 5 a^{8} + 5 a^{7} + 4 a^{6} + a^{5} + 25 a^{4} + 14 a^{3} + 11 a^{2} + 17 a + 27\right)\cdot 29 + \left(17 a^{9} + 26 a^{8} + 4 a^{7} + 9 a^{6} + 9 a^{5} + 27 a^{4} + 11 a^{3} + 23 a^{2} + 24 a + 9\right)\cdot 29^{2} + \left(21 a^{9} + 12 a^{8} + 28 a^{7} + 11 a^{6} + a^{5} + 13 a^{4} + 18 a^{3} + a^{2} + 24 a + 20\right)\cdot 29^{3} + \left(13 a^{9} + 2 a^{8} + 14 a^{7} + 2 a^{6} + 2 a^{5} + 11 a^{4} + 25 a^{3} + 20 a^{2} + 25 a + 8\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 16 }$ $=$ \( 3 a^{9} + 20 a^{8} + 11 a^{7} + 28 a^{6} + 28 a^{5} + 9 a^{4} + 6 a^{3} + 18 a^{2} + 20 a + 20 + \left(2 a^{9} + 24 a^{8} + a^{7} + 13 a^{6} + 15 a^{5} + 6 a^{4} + 21 a^{3} + 9 a^{2} + 24 a + 13\right)\cdot 29 + \left(17 a^{9} + 10 a^{8} + 21 a^{7} + 13 a^{6} + 2 a^{5} + 25 a^{4} + 24 a^{3} + 23 a^{2} + 19 a + 22\right)\cdot 29^{2} + \left(a^{9} + 4 a^{8} + 19 a^{7} + 21 a^{6} + 5 a^{5} + 15 a^{4} + 10 a^{3} + 14 a^{2} + 8 a + 28\right)\cdot 29^{3} + \left(17 a^{9} + 26 a^{8} + 13 a^{7} + 28 a^{6} + 25 a^{5} + 18 a^{4} + 10 a^{3} + 17 a^{2} + a + 4\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 17 }$ $=$ \( 25 a^{9} + 21 a^{8} + 3 a^{7} + 10 a^{6} + 20 a^{5} + 3 a^{4} + 3 a^{3} + 25 a^{2} + 6 a + 20 + \left(2 a^{9} + 28 a^{8} + 5 a^{7} + 11 a^{6} + 13 a^{5} + 19 a^{4} + 27 a^{3} + 17 a^{2} + 16 a + 1\right)\cdot 29 + \left(14 a^{9} + 8 a^{8} + 9 a^{7} + a^{6} + 3 a^{5} + 25 a^{4} + 15 a^{3} + 14 a^{2} + 28 a + 3\right)\cdot 29^{2} + \left(27 a^{9} + 14 a^{8} + 12 a^{7} + 6 a^{6} + 16 a^{5} + 28 a^{4} + 5 a^{3} + 5 a^{2} + 28 a + 27\right)\cdot 29^{3} + \left(a^{9} + 4 a^{8} + 4 a^{7} + 16 a^{6} + 12 a^{5} + 5 a^{4} + 8 a^{3} + 11 a^{2} + 21 a + 28\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 18 }$ $=$ \( 26 a^{9} + 6 a^{8} + 10 a^{7} + 11 a^{6} + 26 a^{4} + 24 a^{3} + 6 a^{2} + 19 a + 9 + \left(20 a^{9} + 2 a^{8} + 23 a^{7} + 9 a^{6} + 8 a^{5} + 12 a^{4} + a^{3} + 11 a^{2} + 15 a + 23\right)\cdot 29 + \left(22 a^{9} + 16 a^{8} + 2 a^{7} + 2 a^{6} + 2 a^{5} + 5 a^{4} + 23 a^{3} + 21 a^{2} + 19 a + 12\right)\cdot 29^{2} + \left(26 a^{9} + 22 a^{8} + 24 a^{7} + 16 a^{6} + 11 a^{5} + 14 a^{4} + 22 a^{3} + 16 a^{2} + 4 a + 10\right)\cdot 29^{3} + \left(4 a^{9} + 4 a^{8} + 3 a^{6} + 2 a^{5} + 27 a^{4} + 14 a^{3} + 20 a^{2} + 12 a + 25\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 19 }$ $=$ \( 26 a^{8} + 5 a^{7} + 23 a^{6} + 22 a^{5} + 17 a^{4} + 28 a^{3} + 28 a^{2} + 26 a + 16 + \left(28 a^{9} + 24 a^{8} + 17 a^{7} + a^{6} + 2 a^{5} + 26 a^{4} + 17 a^{3} + 18 a^{2} + 10 a + 3\right)\cdot 29 + \left(7 a^{9} + 9 a^{8} + 23 a^{7} + a^{5} + 22 a^{4} + 20 a^{3} + 16 a^{2} + 15 a + 7\right)\cdot 29^{2} + \left(15 a^{9} + 8 a^{8} + 26 a^{7} + 18 a^{6} + 22 a^{5} + 4 a^{4} + 24 a^{3} + 26 a^{2} + 3 a + 25\right)\cdot 29^{3} + \left(16 a^{9} + 19 a^{8} + 11 a^{7} + 20 a^{6} + 6 a^{4} + 18 a^{3} + 8 a^{2} + 12 a + 13\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 20 }$ $=$ \( 13 a^{9} + 17 a^{8} + 21 a^{7} + 4 a^{6} + 27 a^{5} + 7 a^{4} + 14 a^{3} + 7 a^{2} + 7 a + 6 + \left(5 a^{9} + 15 a^{8} + 14 a^{7} + 23 a^{6} + 12 a^{5} + 14 a^{4} + 3 a^{3} + 20 a^{2} + 13 a + 5\right)\cdot 29 + \left(a^{8} + 10 a^{7} + 19 a^{6} + 12 a^{5} + 3 a^{4} + 24 a^{3} + 5 a^{2} + 8 a + 7\right)\cdot 29^{2} + \left(a^{9} + 5 a^{8} + 23 a^{7} + 2 a^{6} + 9 a^{5} + 18 a^{4} + 18 a^{3} + 3 a^{2} + 14 a + 16\right)\cdot 29^{3} + \left(27 a^{9} + 11 a^{8} + 25 a^{7} + 17 a^{6} + 17 a^{5} + 24 a^{4} + 28 a^{3} + 16 a^{2} + 24 a + 16\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

SizeOrderAction on $r_1, \ldots, r_{ 20 }$ Character value
$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$$4$$(1,4,18,15)(2,3,17,16)(5,12,14,7)(6,11,13,8)(9,20,10,19)$$-\zeta_{20}^{5}$
$1$$4$$(1,15,18,4)(2,16,17,3)(5,7,14,12)(6,8,13,11)(9,19,10,20)$$\zeta_{20}^{5}$
$1$$5$$(1,17,13,9,5)(2,6,10,14,18)(3,11,19,7,15)(4,16,8,20,12)$$-\zeta_{20}^{2}$
$1$$5$$(1,13,5,17,9)(2,10,18,6,14)(3,19,15,11,7)(4,8,12,16,20)$$\zeta_{20}^{4}$
$1$$5$$(1,9,17,5,13)(2,14,6,18,10)(3,7,11,15,19)(4,20,16,12,8)$$-\zeta_{20}^{6}$
$1$$5$$(1,5,9,13,17)(2,18,14,10,6)(3,15,7,19,11)(4,12,20,8,16)$$\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1$
$1$$10$$(1,10,17,14,13,18,9,2,5,6)(3,12,11,4,19,16,7,8,15,20)$$\zeta_{20}^{6}$
$1$$10$$(1,14,9,6,17,18,5,10,13,2)(3,4,7,20,11,16,15,12,19,8)$$-\zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1$
$1$$10$$(1,2,13,10,5,18,17,6,9,14)(3,8,19,12,15,16,11,20,7,4)$$\zeta_{20}^{2}$
$1$$10$$(1,6,5,2,9,18,13,14,17,10)(3,20,15,8,7,16,19,4,11,12)$$-\zeta_{20}^{4}$
$1$$20$$(1,7,10,8,17,15,14,20,13,3,18,12,9,11,2,4,5,19,6,16)$$-\zeta_{20}^{3}$
$1$$20$$(1,8,14,3,9,4,6,7,17,20,18,11,5,16,10,15,13,12,2,19)$$-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$
$1$$20$$(1,20,2,7,13,4,10,3,5,8,18,19,17,12,6,15,9,16,14,11)$$-\zeta_{20}$
$1$$20$$(1,3,6,20,5,15,2,8,9,7,18,16,13,19,14,4,17,11,10,12)$$-\zeta_{20}^{7}$
$1$$20$$(1,12,10,11,17,4,14,19,13,16,18,7,9,8,2,15,5,20,6,3)$$\zeta_{20}^{3}$
$1$$20$$(1,11,14,16,9,15,6,12,17,19,18,8,5,3,10,4,13,7,2,20)$$\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$
$1$$20$$(1,19,2,12,13,15,10,16,5,11,18,20,17,7,6,4,9,3,14,8)$$\zeta_{20}$
$1$$20$$(1,16,6,19,5,4,2,11,9,12,18,3,13,20,14,15,17,8,10,7)$$\zeta_{20}^{7}$

The blue line marks the conjugacy class containing complex conjugation.