Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(526731062976\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 673^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.2461736148492.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Projective image: | $S_4\wr C_2$ |
| Projective field: | Galois closure of 8.2.2461736148492.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a^{2} + 4 a + \left(12 a^{2} + 10 a + 4\right)\cdot 13 + \left(2 a^{2} + 12\right)\cdot 13^{2} + \left(a^{2} + 4 a + 12\right)\cdot 13^{3} + \left(8 a^{2} + 4 a + 7\right)\cdot 13^{4} + \left(5 a^{2} + 2 a + 2\right)\cdot 13^{5} + \left(10 a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(6 a^{2} + 5 a\right)\cdot 13^{7} + \left(12 a^{2} + 11 a + 10\right)\cdot 13^{8} + \left(3 a^{2} + 11 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a^{2} + 9 a + 1 + \left(3 a^{2} + 6 a + 12\right)\cdot 13 + \left(7 a^{2} + 11 a + 9\right)\cdot 13^{2} + \left(2 a^{2} + 4 a + 12\right)\cdot 13^{3} + \left(7 a^{2} + 4 a\right)\cdot 13^{4} + \left(12 a^{2} + 4 a + 6\right)\cdot 13^{5} + \left(10 a^{2} + 10 a + 10\right)\cdot 13^{6} + \left(10 a^{2} + 7 a + 8\right)\cdot 13^{7} + \left(a^{2} + 10 a + 8\right)\cdot 13^{8} + \left(2 a^{2} + 3 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a^{2} + 5 a + 5 + \left(11 a^{2} + 5 a + 6\right)\cdot 13 + \left(7 a^{2} + a + 1\right)\cdot 13^{2} + \left(9 a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(11 a^{2} + 10 a + 12\right)\cdot 13^{4} + \left(4 a^{2} + 11 a + 5\right)\cdot 13^{5} + \left(10 a^{2} + 6 a + 10\right)\cdot 13^{6} + \left(9 a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(10 a^{2} + 11 a + 7\right)\cdot 13^{8} + \left(a^{2} + a + 12\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 3\cdot 13 + 12\cdot 13^{2} + 10\cdot 13^{3} + 12\cdot 13^{4} + 5\cdot 13^{5} + 12\cdot 13^{6} + 3\cdot 13^{7} + 7\cdot 13^{8} + 11\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 4 a + 3 + \left(a^{2} + 10 a + 2\right)\cdot 13 + \left(2 a^{2} + 10 a + 11\right)\cdot 13^{2} + \left(2 a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(6 a^{2} + 10 a + 9\right)\cdot 13^{4} + \left(2 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(5 a^{2} + 2 a + 12\right)\cdot 13^{6} + \left(9 a^{2} + 12 a + 3\right)\cdot 13^{7} + \left(2 a^{2} + 2 a + 1\right)\cdot 13^{8} + \left(7 a^{2} + 12 a + 11\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a^{2} + 11 a + 7 + \left(a^{2} + 8 a\right)\cdot 13 + \left(6 a^{2} + 10 a + 4\right)\cdot 13^{2} + \left(6 a^{2} + 6 a + 9\right)\cdot 13^{3} + \left(11 a^{2} + 10 a + 6\right)\cdot 13^{4} + \left(12 a^{2} + 10 a + 6\right)\cdot 13^{5} + \left(3 a^{2} + 4 a + 5\right)\cdot 13^{6} + \left(a^{2} + 2 a\right)\cdot 13^{7} + \left(3 a^{2} + 7 a + 6\right)\cdot 13^{8} + \left(10 a^{2} + 9 a + 5\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a^{2} + 6 a + 1 + \left(8 a^{2} + 10 a + 10\right)\cdot 13 + \left(12 a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(3 a^{2} + a + 5\right)\cdot 13^{3} + \left(7 a^{2} + 11 a + 5\right)\cdot 13^{4} + \left(10 a + 7\right)\cdot 13^{5} + \left(11 a^{2} + 10 a + 10\right)\cdot 13^{6} + \left(2 a + 12\right)\cdot 13^{7} + \left(8 a^{2} + 8 a + 3\right)\cdot 13^{8} + \left(12 a + 1\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 + 13^{2} + 5\cdot 13^{3} + 8\cdot 13^{4} + 13^{5} + 10\cdot 13^{6} + 12\cdot 13^{7} + 6\cdot 13^{8} + 8\cdot 13^{9} +O(13^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,6)(4,7)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $1$ |
| $12$ | $2$ | $(1,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $1$ |
| $36$ | $2$ | $(1,3)(2,6)(4,7)$ | $-1$ |
| $16$ | $3$ | $(1,5,8)$ | $0$ |
| $64$ | $3$ | $(1,5,8)(4,6,7)$ | $0$ |
| $12$ | $4$ | $(2,4,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,3,5,8)(2,4,6,7)$ | $1$ |
| $36$ | $4$ | $(1,3,5,8)(2,6)(4,7)$ | $1$ |
| $72$ | $4$ | $(1,2,5,6)(3,4,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,3)(2,4,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,8,5)(2,6)(4,7)$ | $0$ |
| $96$ | $6$ | $(1,3)(4,7,6)$ | $0$ |
| $192$ | $6$ | $(1,4,5,6,8,7)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,4,5,6,8,7)$ | $-1$ |
| $96$ | $12$ | $(1,5,8)(2,4,6,7)$ | $0$ |