Basic invariants
| Dimension: | $5$ |
| Group: | $A_5\times C_2$ |
| Conductor: | \(9037905787\)\(\medspace = 2083^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 10.0.39214470002250643.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T75 |
| Parity: | odd |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.4338889.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{5} + x + 14 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a^{4} + 2 a^{3} + 5 a^{2} + 16 a + 10 + \left(13 a^{4} + 13 a^{3} + 9 a^{2} + 4 a + 10\right)\cdot 17 + \left(12 a^{4} + 13 a^{3} + 15 a^{2} + 15 a + 13\right)\cdot 17^{2} + \left(3 a^{4} + 11 a^{3} + 9 a^{2} + 3 a + 9\right)\cdot 17^{3} + \left(4 a^{4} + 5 a^{3} + 14 a^{2} + 5 a + 13\right)\cdot 17^{4} + \left(10 a^{4} + 8 a^{3} + 8 a^{2} + 15 a + 4\right)\cdot 17^{5} + \left(5 a^{4} + 3 a^{3} + 6 a + 11\right)\cdot 17^{6} + \left(15 a^{4} + 11 a^{3} + 7 a^{2} + 16 a + 15\right)\cdot 17^{7} + \left(16 a^{4} + 9 a^{3} + 16 a^{2} + 2 a + 16\right)\cdot 17^{8} + \left(16 a^{4} + 6 a^{3} + 9 a^{2} + 2 a + 16\right)\cdot 17^{9} +O(17^{10})\)
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| $r_{ 2 }$ | $=$ |
\( 4 a^{4} + 5 a^{3} + 10 a^{2} + 7 a + 10 + \left(13 a^{3} + 13 a^{2} + 13 a + 3\right)\cdot 17 + \left(10 a^{4} + 2 a^{3} + 2 a^{2} + 9 a + 1\right)\cdot 17^{2} + \left(14 a^{4} + 2 a^{2} + 14 a + 15\right)\cdot 17^{3} + \left(9 a^{4} + 14 a^{3} + a^{2} + 11 a + 7\right)\cdot 17^{4} + \left(7 a^{4} + 2 a^{3} + 3 a^{2} + 12 a + 9\right)\cdot 17^{5} + \left(15 a^{3} + 6 a^{2} + 11 a + 10\right)\cdot 17^{6} + \left(3 a^{4} + 9 a^{3} + 3 a^{2} + 2 a + 12\right)\cdot 17^{7} + \left(9 a^{4} + 11 a^{3} + 15 a^{2} + 10\right)\cdot 17^{8} + \left(8 a^{4} + 15 a^{2} + a + 13\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a^{4} + 16 a^{3} + 14 a^{2} + 9 a + 4 + \left(2 a^{4} + 4 a^{3} + 8 a^{2} + 4 a + 5\right)\cdot 17 + \left(8 a^{4} + 13 a^{3} + 8 a^{2} + 16 a + 13\right)\cdot 17^{2} + \left(6 a^{4} + 16 a^{3} + 15 a^{2} + 8 a + 1\right)\cdot 17^{3} + \left(9 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 4\right)\cdot 17^{4} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + 6 a + 10\right)\cdot 17^{5} + \left(16 a^{4} + 4 a^{3} + 16 a^{2} + a + 16\right)\cdot 17^{6} + \left(4 a^{4} + 11 a^{3} + 4 a^{2} + 16 a + 3\right)\cdot 17^{7} + \left(8 a^{4} + 13 a^{3} + 5 a^{2} + 4 a + 3\right)\cdot 17^{8} + \left(11 a^{4} + 3 a^{3} + 4 a^{2} + 11 a + 9\right)\cdot 17^{9} +O(17^{10})\)
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| $r_{ 4 }$ | $=$ |
\( 6 a^{4} + 8 a^{3} + 13 a^{2} + 6 a + 15 + \left(13 a^{4} + 12 a^{3} + 12 a^{2} + 8 a + 3\right)\cdot 17 + \left(4 a^{4} + 15 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 17^{2} + \left(13 a^{4} + 12 a^{3} + 9 a^{2} + 14 a\right)\cdot 17^{3} + \left(11 a^{4} + 13 a^{3} + 13 a^{2} + 7 a + 6\right)\cdot 17^{4} + \left(7 a^{3} + 12 a + 7\right)\cdot 17^{5} + \left(11 a^{4} + 10 a^{3} + 16 a^{2} + 11 a + 5\right)\cdot 17^{6} + \left(12 a^{4} + 2 a^{3} + 16 a^{2} + 2 a + 3\right)\cdot 17^{7} + \left(8 a^{4} + 8 a^{3} + 6 a^{2} + 16 a\right)\cdot 17^{8} + \left(14 a^{4} + 9 a^{3} + 13 a^{2} + 13 a + 15\right)\cdot 17^{9} +O(17^{10})\)
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| $r_{ 5 }$ | $=$ |
\( 8 a^{4} + 6 a^{3} + 9 a^{2} + 8 a + 3 + \left(12 a^{4} + 9 a^{3} + 16 a^{2} + 3 a + 3\right)\cdot 17 + \left(12 a^{4} + 2 a^{3} + 12 a^{2} + 14 a + 10\right)\cdot 17^{2} + \left(16 a^{4} + 3 a^{3} + 2 a^{2} + 6 a + 6\right)\cdot 17^{3} + \left(3 a^{4} + 10 a^{3} + 2 a^{2} + 5 a + 13\right)\cdot 17^{4} + \left(9 a^{4} + 4 a^{3} + a^{2} + 8 a + 10\right)\cdot 17^{5} + \left(7 a^{4} + 6 a^{3} + 6 a^{2} + 7 a + 2\right)\cdot 17^{6} + \left(7 a^{4} + 11 a^{3} + 7 a^{2} + 6 a + 16\right)\cdot 17^{7} + \left(7 a^{4} + 6 a^{3} + 15 a^{2} + 6 a + 5\right)\cdot 17^{8} + \left(16 a^{4} + 7 a^{3} + 12 a^{2} + 8 a + 6\right)\cdot 17^{9} +O(17^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 9 a^{4} + 12 a^{3} + 6 a^{2} + 14 + \left(9 a^{4} + 16 a^{3} + 11 a^{2} + a\right)\cdot 17 + \left(4 a^{4} + 9 a^{3} + 12 a + 7\right)\cdot 17^{2} + \left(12 a^{4} + 6 a^{3} + 8 a^{2} + 8 a + 6\right)\cdot 17^{3} + \left(16 a^{4} + 7 a^{2} + 8 a + 13\right)\cdot 17^{4} + \left(12 a^{3} + 4 a^{2} + 8 a\right)\cdot 17^{5} + \left(11 a^{4} + a^{3} + 9 a^{2} + a + 2\right)\cdot 17^{6} + \left(13 a^{4} + 16 a^{3} + 15 a^{2} + 14 a + 4\right)\cdot 17^{7} + \left(13 a^{4} + 14 a^{3} + 13 a^{2} + 6 a + 4\right)\cdot 17^{8} + \left(7 a^{3} + 15 a^{2} + 3 a + 14\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 a^{4} + a^{3} + a^{2} + 10 a + 8 + \left(14 a^{4} + 3 a^{3} + 3 a^{2} + 15 a + 1\right)\cdot 17 + \left(a^{4} + 13 a^{3} + 12 a^{2} + 11 a + 15\right)\cdot 17^{2} + \left(11 a^{4} + 11 a^{2} + 8\right)\cdot 17^{3} + \left(15 a^{4} + 7 a^{3} + 3 a + 12\right)\cdot 17^{4} + \left(7 a^{4} + 12 a^{3} + 11 a^{2} + 6 a + 9\right)\cdot 17^{5} + \left(11 a^{4} + 9 a^{3} + 8 a^{2} + 16 a + 12\right)\cdot 17^{6} + \left(15 a^{4} + 15 a^{3} + 4 a^{2} + 16 a + 5\right)\cdot 17^{7} + \left(10 a^{4} + 16 a^{3} + 16 a^{2} + 7 a + 5\right)\cdot 17^{8} + \left(8 a^{4} + 4 a^{3} + 11 a + 10\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{4} + 16 a^{3} + 16 a^{2} + 2 a + 13 + \left(8 a^{4} + 2 a^{3} + 7 a^{2} + 10 a + 13\right)\cdot 17 + \left(15 a^{4} + 11 a^{3} + 6 a^{2} + 14 a + 15\right)\cdot 17^{2} + \left(13 a^{4} + 15 a^{3} + 15 a^{2} + 14 a\right)\cdot 17^{3} + \left(10 a^{4} + 2 a^{3} + 16 a^{2} + 3 a + 12\right)\cdot 17^{4} + \left(6 a^{4} + 2 a^{3} + 14 a^{2} + 8 a + 8\right)\cdot 17^{5} + \left(9 a^{3} + a^{2} + 12 a + 10\right)\cdot 17^{6} + \left(14 a^{4} + 2 a^{3} + 3 a^{2} + a + 14\right)\cdot 17^{7} + \left(2 a^{4} + a^{3} + 2 a + 15\right)\cdot 17^{8} + \left(13 a^{4} + 15 a^{3} + 5 a^{2} + 16 a + 13\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 13 a^{4} + 15 a^{3} + 14 a^{2} + 7 a + 7 + \left(4 a^{4} + 10 a^{3} + 9 a^{2}\right)\cdot 17 + \left(5 a^{4} + 10 a^{3} + 4 a^{2} + 14 a + 11\right)\cdot 17^{2} + \left(14 a^{4} + 11 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 17^{3} + \left(14 a^{4} + 14 a^{3} + 5 a^{2} + 4 a + 15\right)\cdot 17^{4} + \left(16 a^{4} + 10 a^{3} + 16 a^{2} + 4 a + 16\right)\cdot 17^{5} + \left(11 a^{4} + 13 a^{3} + 13 a^{2} + 9 a + 12\right)\cdot 17^{6} + \left(15 a^{4} + 2 a^{3} + 2 a^{2} + 9 a + 5\right)\cdot 17^{7} + \left(7 a^{4} + 3 a^{3} + 12 a^{2} + a + 6\right)\cdot 17^{8} + \left(5 a^{4} + 7 a^{3} + 15 a^{2} + 13 a + 4\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 10 }$ | $=$ |
\( 14 a^{4} + 4 a^{3} + 14 a^{2} + 3 a + 1 + \left(5 a^{4} + 15 a^{3} + 8 a^{2} + 6 a + 8\right)\cdot 17 + \left(9 a^{4} + 8 a^{3} + 9 a^{2} + 8 a + 7\right)\cdot 17^{2} + \left(12 a^{4} + 5 a^{3} + 16 a^{2} + 9 a + 13\right)\cdot 17^{3} + \left(4 a^{4} + 5 a^{3} + 11 a^{2} + 13 a + 3\right)\cdot 17^{4} + \left(16 a^{4} + 15 a^{3} + 4 a^{2} + 2 a + 6\right)\cdot 17^{5} + \left(8 a^{4} + 10 a^{3} + 6 a^{2} + 6 a\right)\cdot 17^{6} + \left(16 a^{4} + a^{3} + 2 a^{2} + 15 a + 3\right)\cdot 17^{7} + \left(15 a^{4} + 16 a^{3} + a + 16\right)\cdot 17^{8} + \left(5 a^{4} + 4 a^{3} + 8 a^{2} + 4 a + 14\right)\cdot 17^{9} +O(17^{10})\)
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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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $1$ | $2$ | $(1,5)(2,10)(3,7)(4,6)(8,9)$ | $-5$ |
| $15$ | $2$ | $(1,9)(2,4)(3,7)(5,8)(6,10)$ | $-1$ |
| $15$ | $2$ | $(1,8)(2,6)(4,10)(5,9)$ | $1$ |
| $20$ | $3$ | $(1,6,8)(4,9,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,8,2,6)(4,5,7,9,10)$ | $0$ |
| $12$ | $5$ | $(1,2,3,6,8)(4,9,5,10,7)$ | $0$ |
| $20$ | $6$ | $(1,9,6,5,8,4)(2,10)(3,7)$ | $1$ |
| $12$ | $10$ | $(1,10,3,4,8,5,2,7,6,9)$ | $0$ |
| $12$ | $10$ | $(1,4,2,9,3,5,6,10,8,7)$ | $0$ |