Basic invariants
Dimension: | $24$ |
Group: | $A_5 \wr C_2$ |
Conductor: | \(382\!\cdots\!000\)\(\medspace = 2^{54} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{14} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 10.10.23300317255158046392320000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 120 |
Parity: | even |
Projective image: | $A_5 \wr C_2$ |
Projective field: | Galois closure of 10.10.23300317255158046392320000.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{4} + 2x^{2} + 15x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 20 a^{2} + 8 a + 1 + \left(6 a^{3} + 20 a^{2} + 25 a + 24\right)\cdot 29 + \left(21 a^{3} + 10 a^{2} + 9 a + 5\right)\cdot 29^{2} + \left(10 a^{3} + 20 a^{2} + 17 a + 28\right)\cdot 29^{3} + \left(8 a^{3} + 18 a^{2} + 5 a + 5\right)\cdot 29^{4} + \left(9 a^{3} + 7 a^{2} + 10 a + 18\right)\cdot 29^{5} + \left(a^{2} + 16 a + 3\right)\cdot 29^{6} + \left(4 a^{3} + 5 a^{2} + 20 a + 14\right)\cdot 29^{7} + \left(27 a^{3} + 2 a + 19\right)\cdot 29^{8} + \left(16 a^{3} + 4 a^{2} + 22 a + 15\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 2 }$ |
$=$ |
\( 24 a^{3} + 2 a^{2} + 17 a + 21 + \left(3 a^{3} + 27 a^{2} + 8 a + 10\right)\cdot 29 + \left(3 a^{3} + 3 a^{2} + 25 a + 2\right)\cdot 29^{2} + \left(23 a^{3} + 20 a^{2} + 10 a + 19\right)\cdot 29^{3} + \left(22 a^{3} + 22 a^{2} + 23 a + 10\right)\cdot 29^{4} + \left(23 a^{3} + 14 a^{2} + 23 a + 15\right)\cdot 29^{5} + \left(10 a^{3} + 22 a^{2} + 26 a + 3\right)\cdot 29^{6} + \left(16 a^{3} + 6 a^{2} + 18 a + 17\right)\cdot 29^{7} + \left(6 a^{3} + 4 a^{2} + 6 a + 8\right)\cdot 29^{8} + \left(10 a^{3} + 28 a^{2} + 6 a\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 13 a^{3} + 27 a^{2} + 2 + \left(4 a^{3} + 17 a^{2} + 6 a + 11\right)\cdot 29 + \left(11 a^{3} + 17 a^{2} + 2 a + 8\right)\cdot 29^{2} + \left(13 a^{3} + 10 a^{2} + 4 a + 19\right)\cdot 29^{3} + \left(28 a^{3} + 22 a^{2} + 6 a + 3\right)\cdot 29^{4} + \left(16 a^{3} + 2 a^{2} + 15 a + 20\right)\cdot 29^{5} + \left(8 a^{3} + 8 a^{2} + 8 a + 23\right)\cdot 29^{6} + \left(2 a^{3} + 5 a^{2} + 12 a + 16\right)\cdot 29^{7} + \left(21 a^{3} + 22 a^{2} + 14 a + 9\right)\cdot 29^{8} + \left(17 a^{3} + 23 a^{2} + 10 a + 15\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 19 a^{3} + 22 a^{2} + 12 a + 21 + \left(23 a^{3} + 10 a^{2} + 8 a + 9\right)\cdot 29 + \left(22 a^{3} + 18 a^{2} + 7 a + 24\right)\cdot 29^{2} + \left(16 a^{3} + 14 a^{2} + 22 a + 10\right)\cdot 29^{3} + \left(14 a^{3} + 20 a^{2} + 25 a + 12\right)\cdot 29^{4} + \left(5 a^{3} + 7 a^{2} + 3 a + 26\right)\cdot 29^{5} + \left(24 a^{3} + 11 a^{2} + 10 a + 13\right)\cdot 29^{6} + \left(24 a^{3} + 11 a^{2} + 20 a + 15\right)\cdot 29^{7} + \left(26 a^{3} + 2 a^{2} + a + 18\right)\cdot 29^{8} + \left(14 a^{3} + 2 a^{2} + 6 a + 5\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 26 a^{3} + 18 a^{2} + 9 a + 16 + \left(23 a^{3} + 8 a^{2} + 18 a + 17\right)\cdot 29 + \left(2 a^{3} + 11 a^{2} + 9 a + 2\right)\cdot 29^{2} + \left(17 a^{3} + 12 a^{2} + 14 a + 12\right)\cdot 29^{3} + \left(6 a^{3} + 25 a^{2} + 20 a + 14\right)\cdot 29^{4} + \left(26 a^{3} + 10 a^{2} + 28 a + 23\right)\cdot 29^{5} + \left(24 a^{3} + 8 a^{2} + 22 a + 11\right)\cdot 29^{6} + \left(26 a^{3} + 7 a^{2} + 4 a + 12\right)\cdot 29^{7} + \left(11 a^{3} + 4 a^{2} + 10 a + 26\right)\cdot 29^{8} + \left(8 a^{3} + 28 a^{2} + 19 a + 1\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 23 a^{3} + 21 a^{2} + 10 a + 26 + \left(18 a^{3} + 15 a^{2} + 22 a + 12\right)\cdot 29 + \left(12 a^{3} + 28 a^{2} + 24 a + 19\right)\cdot 29^{2} + \left(2 a^{3} + 28 a^{2} + 2 a + 20\right)\cdot 29^{3} + \left(5 a^{3} + 17 a^{2} + 24 a + 22\right)\cdot 29^{4} + \left(13 a^{3} + 13 a^{2} + 9 a + 26\right)\cdot 29^{5} + \left(18 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 29^{6} + \left(20 a^{3} + 22 a^{2} + 9 a + 22\right)\cdot 29^{7} + \left(14 a^{3} + 17 a^{2} + 11 a + 3\right)\cdot 29^{8} + \left(22 a^{3} + 25 a^{2} + 6 a + 7\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 5 a^{3} + 27 a^{2} + 12 a + 28 + \left(25 a^{3} + a^{2} + 20 a\right)\cdot 29 + \left(25 a^{3} + 25 a^{2} + 3 a + 11\right)\cdot 29^{2} + \left(5 a^{3} + 8 a^{2} + 18 a + 24\right)\cdot 29^{3} + \left(6 a^{3} + 6 a^{2} + 5 a + 17\right)\cdot 29^{4} + \left(5 a^{3} + 14 a^{2} + 5 a + 1\right)\cdot 29^{5} + \left(18 a^{3} + 6 a^{2} + 2 a + 5\right)\cdot 29^{6} + \left(12 a^{3} + 22 a^{2} + 10 a + 12\right)\cdot 29^{7} + \left(22 a^{3} + 24 a^{2} + 22 a + 26\right)\cdot 29^{8} + \left(18 a^{3} + 22 a + 3\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 3 a^{3} + 15 a^{2} + 6 a + 27 + \left(5 a^{3} + 25 a^{2} + 24 a + 20\right)\cdot 29 + \left(21 a^{3} + 12 a^{2} + 28 a + 12\right)\cdot 29^{2} + \left(7 a^{3} + 5 a^{2} + 8 a + 20\right)\cdot 29^{3} + \left(28 a^{3} + 20 a^{2} + 27 a + 24\right)\cdot 29^{4} + \left(23 a^{3} + 8 a^{2} + 12\right)\cdot 29^{5} + \left(28 a^{3} + 21 a^{2} + 15 a + 18\right)\cdot 29^{6} + \left(4 a^{3} + 23 a + 27\right)\cdot 29^{7} + \left(13 a^{3} + 19 a^{2} + 15 a + 23\right)\cdot 29^{8} + \left(10 a^{3} + 17 a^{2} + 4 a + 15\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 26 a^{3} + 5 a^{2} + 7 a + 22 + \left(9 a^{3} + 28 a^{2} + 12 a + 26\right)\cdot 29 + \left(22 a^{3} + 8 a^{2} + 15 a + 21\right)\cdot 29^{2} + \left(14 a^{3} + 6 a^{2} + 23 a + 20\right)\cdot 29^{3} + \left(7 a^{3} + 17 a^{2} + 9 a + 5\right)\cdot 29^{4} + \left(9 a^{3} + 24 a^{2} + 4 a + 8\right)\cdot 29^{5} + \left(3 a^{3} + 26 a^{2} + 28 a + 5\right)\cdot 29^{6} + \left(13 a^{3} + 16 a^{2} + 16 a + 19\right)\cdot 29^{7} + \left(2 a^{3} + 16 a^{2} + 21 a + 9\right)\cdot 29^{8} + \left(8 a^{3} + 10 a^{2} + 2 a + 18\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 10 }$ |
$=$ |
\( 6 a^{3} + 17 a^{2} + 6 a + 12 + \left(24 a^{3} + 17 a^{2} + 28 a + 10\right)\cdot 29 + \left(a^{3} + 7 a^{2} + 17 a + 7\right)\cdot 29^{2} + \left(4 a^{3} + 17 a^{2} + 22 a + 27\right)\cdot 29^{3} + \left(17 a^{3} + 2 a^{2} + 25 a + 26\right)\cdot 29^{4} + \left(11 a^{3} + 11 a^{2} + 13 a + 20\right)\cdot 29^{5} + \left(7 a^{3} + 27 a^{2} + 12 a + 22\right)\cdot 29^{6} + \left(19 a^{3} + 17 a^{2} + 8 a + 16\right)\cdot 29^{7} + \left(27 a^{3} + 4 a^{2} + 9 a + 27\right)\cdot 29^{8} + \left(16 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 29^{9} +O(29^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $24$ | $24$ |
$30$ | $2$ | $(4,5)(7,9)$ | $-4$ | $-4$ |
$60$ | $2$ | $(1,4)(2,5)(3,10)(6,9)(7,8)$ | $0$ | $0$ |
$225$ | $2$ | $(1,2)(4,5)(6,8)(7,9)$ | $0$ | $0$ |
$40$ | $3$ | $(4,5,7)$ | $3$ | $3$ |
$400$ | $3$ | $(1,2,8)(4,5,7)$ | $0$ | $0$ |
$900$ | $4$ | $(1,7,8,4)(2,10,3,5)(6,9)$ | $0$ | $0$ |
$24$ | $5$ | $(4,5,7,9,10)$ | $4 \zeta_{5}^{3} + 4 \zeta_{5}^{2} + 1$ | $-4 \zeta_{5}^{3} - 4 \zeta_{5}^{2} - 3$ |
$24$ | $5$ | $(4,7,9,10,5)$ | $-4 \zeta_{5}^{3} - 4 \zeta_{5}^{2} - 3$ | $4 \zeta_{5}^{3} + 4 \zeta_{5}^{2} + 1$ |
$144$ | $5$ | $(1,2,8,6,3)(4,5,7,9,10)$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2$ | $2 \zeta_{5}^{3} + 2 \zeta_{5}^{2}$ |
$144$ | $5$ | $(1,8,6,3,2)(4,7,9,10,5)$ | $2 \zeta_{5}^{3} + 2 \zeta_{5}^{2}$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2$ |
$288$ | $5$ | $(1,2,8,6,3)(4,7,9,10,5)$ | $-1$ | $-1$ |
$600$ | $6$ | $(1,2)(4,5,7)(6,8)$ | $-1$ | $-1$ |
$1200$ | $6$ | $(1,7,8,5,2,4)(3,10)(6,9)$ | $0$ | $0$ |
$360$ | $10$ | $(1,2)(4,5,7,9,10)(6,8)$ | $1$ | $1$ |
$360$ | $10$ | $(1,2)(4,7,9,10,5)(6,8)$ | $1$ | $1$ |
$720$ | $10$ | $(1,7,8,5,2,10,3,9,6,4)$ | $0$ | $0$ |
$720$ | $10$ | $(1,5,2,10,3,9,6,7,8,4)$ | $0$ | $0$ |
$480$ | $15$ | $(1,2,8)(4,5,7,9,10)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$480$ | $15$ | $(1,2,8)(4,7,9,10,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |