Basic invariants
| Dimension: | $3$ |
| Group: | $A_5\times C_2$ |
| Conductor: | \(3548\)\(\medspace = 2^{2} \cdot 887 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 10.0.8784925479251312.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 12T76 |
| Parity: | odd |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.3147076.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{5} + x + 42 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a^{4} + 17 a^{3} + 5 a^{2} + 24 a + 26 + \left(36 a^{4} + 11 a^{3} + 26 a^{2} + 43 a + 19\right)\cdot 47 + \left(42 a^{4} + 37 a^{3} + 26 a^{2} + 5 a + 15\right)\cdot 47^{2} + \left(34 a^{4} + 40 a^{3} + 8 a^{2} + 11 a + 37\right)\cdot 47^{3} + \left(a^{4} + 25 a^{3} + 10 a^{2} + 11 a + 10\right)\cdot 47^{4} + \left(30 a^{4} + 33 a^{3} + 30 a^{2} + 25 a + 5\right)\cdot 47^{5} + \left(a^{4} + 32 a^{3} + 34 a^{2} + 6 a + 20\right)\cdot 47^{6} + \left(18 a^{4} + 20 a^{3} + 23 a^{2} + 6 a + 14\right)\cdot 47^{7} + \left(38 a^{4} + 45 a^{3} + 8 a^{2} + 21\right)\cdot 47^{8} + \left(23 a^{4} + 32 a^{3} + 45 a^{2} + 37 a + 28\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 2 }$ | $=$ |
\( 16 a^{4} + 7 a^{3} + 22 a^{2} + 41 a + 41 + \left(10 a^{4} + 5 a^{3} + 8 a^{2} + 2 a + 45\right)\cdot 47 + \left(45 a^{4} + 39 a^{3} + 13 a^{2} + 15 a + 7\right)\cdot 47^{2} + \left(39 a^{4} + 20 a^{3} + 11 a^{2} + 11 a + 13\right)\cdot 47^{3} + \left(32 a^{4} + 12 a^{3} + 9 a^{2} + 16 a + 26\right)\cdot 47^{4} + \left(31 a^{4} + 14 a^{3} + 43 a^{2} + 39 a + 6\right)\cdot 47^{5} + \left(36 a^{4} + 30 a^{3} + 32 a + 1\right)\cdot 47^{6} + \left(9 a^{4} + 29 a^{3} + 4 a^{2} + 23 a + 36\right)\cdot 47^{7} + \left(11 a^{4} + 25 a^{3} + 6 a^{2} + 6 a + 27\right)\cdot 47^{8} + \left(7 a^{4} + 38 a^{3} + 5 a^{2} + 5 a + 24\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a^{4} + 43 a^{3} + 15 a^{2} + 43 a + 5 + \left(11 a^{4} + 40 a^{3} + 40 a^{2} + 42 a + 9\right)\cdot 47 + \left(26 a^{4} + 3 a^{3} + 32 a^{2} + 6 a + 2\right)\cdot 47^{2} + \left(33 a^{4} + 23 a^{3} + 11 a^{2} + 8 a + 8\right)\cdot 47^{3} + \left(20 a^{4} + 26 a^{3} + 9 a^{2} + 44 a + 7\right)\cdot 47^{4} + \left(13 a^{4} + 19 a^{3} + 26 a^{2} + 43 a + 20\right)\cdot 47^{5} + \left(4 a^{4} + 37 a^{3} + 23 a^{2} + 32 a + 3\right)\cdot 47^{6} + \left(23 a^{4} + 30 a^{3} + 40 a^{2} + 39 a + 9\right)\cdot 47^{7} + \left(19 a^{4} + 4 a^{3} + 10 a^{2} + 6\right)\cdot 47^{8} + \left(17 a^{4} + 14 a^{3} + 37 a^{2} + 10 a + 42\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a^{4} + 17 a^{3} + 4 a^{2} + 37 a + 45 + \left(32 a^{3} + 29 a^{2} + 35 a + 37\right)\cdot 47 + \left(46 a^{4} + 17 a^{3} + 16 a^{2} + 14 a + 36\right)\cdot 47^{2} + \left(30 a^{4} + a^{3} + 43 a^{2} + 15 a + 24\right)\cdot 47^{3} + \left(19 a^{4} + 42 a^{3} + 13 a^{2} + 12 a + 34\right)\cdot 47^{4} + \left(9 a^{4} + 20 a^{3} + 35 a^{2} + 26 a + 35\right)\cdot 47^{5} + \left(4 a^{4} + 13 a^{3} + 18 a^{2} + 6 a + 12\right)\cdot 47^{6} + \left(12 a^{4} + 24 a^{3} + 6 a^{2} + 20 a\right)\cdot 47^{7} + \left(27 a^{4} + 15 a^{3} + 19 a^{2} + 32 a + 3\right)\cdot 47^{8} + \left(35 a^{4} + 33 a^{3} + 19 a^{2} + 40 a + 19\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a^{4} + 16 a^{3} + 26 a^{2} + 46 a + 38 + \left(16 a^{4} + 21 a^{3} + 5 a^{2} + 2 a + 3\right)\cdot 47 + \left(45 a^{4} + 36 a^{3} + 29 a^{2} + 6 a + 8\right)\cdot 47^{2} + \left(42 a^{4} + 30 a^{3} + 43 a^{2} + 46 a + 6\right)\cdot 47^{3} + \left(36 a^{4} + 40 a^{3} + a + 20\right)\cdot 47^{4} + \left(22 a^{4} + 21 a^{3} + 38 a^{2} + 19 a + 46\right)\cdot 47^{5} + \left(7 a^{4} + 10 a^{3} + 18 a^{2} + 29 a + 5\right)\cdot 47^{6} + \left(a^{4} + 14 a^{3} + 45 a^{2} + 9 a + 29\right)\cdot 47^{7} + \left(13 a^{4} + 14 a^{3} + 46 a^{2} + 4 a + 38\right)\cdot 47^{8} + \left(17 a^{4} + 42 a^{3} + 24 a^{2} + 18 a + 13\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 25 a^{4} + 26 a^{3} + 24 a^{2} + a + 20 + \left(34 a^{4} + 26 a^{3} + 38 a^{2} + 35 a + 46\right)\cdot 47 + \left(5 a^{4} + 30 a^{3} + a^{2} + 9 a + 13\right)\cdot 47^{2} + \left(46 a^{4} + a^{3} + 34 a^{2} + 46 a + 46\right)\cdot 47^{3} + \left(28 a^{4} + 24 a^{3} + 44 a^{2} + 19 a + 41\right)\cdot 47^{4} + \left(40 a^{4} + 7 a^{3} + 35 a^{2} + 45 a + 41\right)\cdot 47^{5} + \left(35 a^{4} + 41 a^{3} + 30 a^{2} + 18 a + 9\right)\cdot 47^{6} + \left(27 a^{3} + 20 a^{2} + 4 a + 38\right)\cdot 47^{7} + \left(8 a^{4} + 19 a^{3} + 22 a^{2} + 30 a + 15\right)\cdot 47^{8} + \left(4 a^{4} + 41 a^{3} + 45 a^{2} + 35 a + 3\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 a^{4} + 44 a^{3} + 45 a^{2} + 14 a + 35 + \left(13 a^{4} + 3 a^{3} + 30 a^{2} + 29 a + 29\right)\cdot 47 + \left(46 a^{4} + 15 a^{3} + 24 a^{2} + 39 a + 27\right)\cdot 47^{2} + \left(17 a^{4} + 23 a^{3} + 11 a^{2} + 46 a + 42\right)\cdot 47^{3} + \left(3 a^{4} + 22 a^{3} + 38 a^{2} + 32 a + 30\right)\cdot 47^{4} + \left(11 a^{4} + 41 a^{3} + 24 a^{2} + 27\right)\cdot 47^{5} + \left(17 a^{4} + 24 a^{3} + 45 a^{2} + 21 a + 32\right)\cdot 47^{6} + \left(45 a^{4} + 8 a^{3} + 40 a^{2} + 38 a + 45\right)\cdot 47^{7} + \left(11 a^{4} + 46 a^{3} + 39 a^{2} + 10 a + 18\right)\cdot 47^{8} + \left(40 a^{4} + 46 a^{3} + 42 a^{2} + 33 a + 13\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 45 a^{4} + 11 a^{3} + 6 a^{2} + 46 a + 36 + \left(33 a^{4} + 37 a^{3} + 2 a^{2} + 44 a + 17\right)\cdot 47 + \left(16 a^{4} + 26 a^{3} + 18 a^{2} + 22 a + 13\right)\cdot 47^{2} + \left(12 a^{4} + 44 a^{3} + 40 a^{2} + 24 a + 19\right)\cdot 47^{3} + \left(21 a^{4} + 25 a^{3} + 34 a^{2} + 31 a + 26\right)\cdot 47^{4} + \left(19 a^{4} + 4 a^{3} + 18 a^{2} + 24 a + 15\right)\cdot 47^{5} + \left(32 a^{4} + 24 a^{3} + 22 a^{2} + 14 a + 35\right)\cdot 47^{6} + \left(12 a^{4} + 2 a^{3} + 32 a^{2} + 38 a\right)\cdot 47^{7} + \left(27 a^{4} + 8 a^{3} + a^{2} + 19 a + 3\right)\cdot 47^{8} + \left(43 a^{4} + 5 a^{3} + 43 a^{2} + 21 a + 16\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 46 a^{4} + 9 a^{3} + 18 + \left(38 a^{4} + 27 a^{3} + 8 a^{2} + 29 a + 12\right)\cdot 47 + \left(5 a^{4} + 20 a^{3} + 46 a^{2} + 35 a + 42\right)\cdot 47^{2} + \left(a^{4} + 37 a^{3} + 35 a^{2} + 32 a\right)\cdot 47^{3} + \left(11 a^{4} + 16 a^{3} + 15 a^{2} + 17 a + 37\right)\cdot 47^{4} + \left(34 a^{4} + 20 a^{3} + 27 a^{2} + 19 a + 36\right)\cdot 47^{5} + \left(42 a^{4} + 42 a^{3} + 4 a^{2} + 41 a + 24\right)\cdot 47^{6} + \left(10 a^{4} + 37 a^{3} + 7 a^{2} + 45 a + 27\right)\cdot 47^{7} + \left(42 a^{4} + 21 a^{3} + 31 a^{2} + 44 a + 5\right)\cdot 47^{8} + \left(30 a^{4} + 8 a^{3} + 33 a^{2} + 7 a + 34\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 10 }$ | $=$ |
\( 46 a^{4} + 45 a^{3} + 41 a^{2} + 30 a + 18 + \left(38 a^{4} + 28 a^{3} + 45 a^{2} + 15 a + 12\right)\cdot 47 + \left(a^{4} + 7 a^{3} + 25 a^{2} + 31 a + 20\right)\cdot 47^{2} + \left(22 a^{4} + 11 a^{3} + 41 a^{2} + 39 a + 36\right)\cdot 47^{3} + \left(11 a^{4} + 45 a^{3} + 10 a^{2} + 46 a + 46\right)\cdot 47^{4} + \left(22 a^{4} + 3 a^{3} + 2 a^{2} + 37 a + 45\right)\cdot 47^{5} + \left(5 a^{4} + 25 a^{3} + 35 a^{2} + 30 a + 41\right)\cdot 47^{6} + \left(7 a^{4} + 38 a^{3} + 13 a^{2} + 8 a + 33\right)\cdot 47^{7} + \left(36 a^{4} + 33 a^{3} + a^{2} + 38 a\right)\cdot 47^{8} + \left(14 a^{4} + 18 a^{3} + 32 a^{2} + 25 a + 40\right)\cdot 47^{9} +O(47^{10})\)
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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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $3$ | $3$ |
| $1$ | $2$ | $(1,6)(2,8)(3,5)(4,9)(7,10)$ | $-3$ | $-3$ |
| $15$ | $2$ | $(1,9)(2,10)(4,6)(7,8)$ | $-1$ | $-1$ |
| $15$ | $2$ | $(1,4)(2,7)(3,5)(6,9)(8,10)$ | $1$ | $1$ |
| $20$ | $3$ | $(1,10,9)(4,6,7)$ | $0$ | $0$ |
| $12$ | $5$ | $(1,2,5,10,9)(3,7,4,6,8)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,5,9,2,10)(3,4,8,7,6)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $20$ | $6$ | $(1,7,9,6,10,4)(2,8)(3,5)$ | $0$ | $0$ |
| $12$ | $10$ | $(1,8,5,7,9,6,2,3,10,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $12$ | $10$ | $(1,7,2,4,5,6,10,8,9,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |