Basic invariants
| Dimension: | $8$ |
| Group: | $((C_3^2:Q_8):C_3):C_2$ |
| Conductor: | \(2824440448203\)\(\medspace = 3^{13} \cdot 11^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.3.2824440448203.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T157 |
| Parity: | odd |
| Projective image: | $C_3^2:\GL(2,3)$ |
| Projective field: | Galois closure of 9.3.2824440448203.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{4} + 2x^{2} + 11x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a^{3} + 7 a^{2} + a + 16 + \left(4 a^{3} + 6 a^{2} + 2 a + 17\right)\cdot 19 + \left(7 a^{3} + 12 a^{2} + 3 a + 18\right)\cdot 19^{2} + \left(2 a^{3} + 9 a + 8\right)\cdot 19^{3} + \left(a^{3} + 5 a^{2} + 14 a + 6\right)\cdot 19^{4} + \left(17 a^{3} + 17 a^{2} + 7 a + 12\right)\cdot 19^{5} + \left(5 a^{3} + 18 a^{2} + 10 a + 13\right)\cdot 19^{6} + \left(17 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 19^{7} + \left(3 a^{3} + 9 a^{2} + 11 a + 4\right)\cdot 19^{8} + \left(16 a^{3} + 18 a^{2} + 15 a + 10\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{3} + 13 a^{2} + 11 a + 11 + \left(15 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 19 + \left(7 a^{3} + 13 a^{2} + 7 a + 15\right)\cdot 19^{2} + \left(9 a^{3} + 3 a^{2} + 7 a + 17\right)\cdot 19^{3} + \left(10 a^{3} + 16 a^{2} + 12 a + 18\right)\cdot 19^{4} + \left(10 a^{3} + 11 a^{2} + 4 a + 14\right)\cdot 19^{5} + \left(16 a^{3} + 15 a^{2} + 3 a + 12\right)\cdot 19^{6} + \left(14 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 19^{7} + \left(18 a^{3} + 12 a^{2} + 3 a + 15\right)\cdot 19^{8} + \left(10 a^{3} + 18 a^{2} + a + 14\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a^{3} + 15 a^{2} + 9 a + 3 + \left(10 a^{3} + 7 a^{2} + 18 a + 3\right)\cdot 19 + \left(18 a^{3} + 10 a^{2} + 3 a + 6\right)\cdot 19^{2} + \left(18 a^{3} + 7 a^{2} + 11 a + 10\right)\cdot 19^{3} + \left(18 a^{3} + 8 a^{2} + 8 a\right)\cdot 19^{4} + \left(9 a^{3} + 4 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(12 a^{3} + 16 a^{2} + 14 a + 18\right)\cdot 19^{6} + \left(13 a^{3} + 11 a^{2} + 14 a + 15\right)\cdot 19^{7} + \left(17 a^{3} + 10 a^{2} + 10 a + 4\right)\cdot 19^{8} + \left(10 a^{3} + 8 a^{2} + 16 a + 4\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 4\cdot 19 + 7\cdot 19^{2} + 17\cdot 19^{4} + 14\cdot 19^{5} + 13\cdot 19^{6} + 7\cdot 19^{7} + 11\cdot 19^{8} + 15\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a^{3} + 2 a^{2} + a + 5 + \left(6 a^{3} + 8 a^{2} + 14 a + 9\right)\cdot 19 + \left(16 a^{3} + 6 a^{2} + 2 a + 16\right)\cdot 19^{2} + \left(10 a^{3} + 12 a^{2} + 12 a + 3\right)\cdot 19^{3} + \left(15 a^{3} + 18 a^{2} + 4 a + 8\right)\cdot 19^{4} + \left(4 a^{3} + 16 a^{2} + 9 a + 13\right)\cdot 19^{5} + \left(7 a^{3} + 8 a + 6\right)\cdot 19^{6} + \left(17 a^{3} + 5 a^{2} + 4 a + 3\right)\cdot 19^{7} + \left(7 a^{3} + 5 a + 15\right)\cdot 19^{8} + \left(11 a^{3} + 15 a^{2} + 4 a + 9\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a^{3} + 13 a^{2} + 15 a + 16 + \left(2 a^{3} + 11 a^{2} + 17 a + 17\right)\cdot 19 + \left(8 a^{3} + 13 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(17 a^{3} + 16 a^{2} + 10 a\right)\cdot 19^{3} + \left(12 a^{3} + a^{2} + 3 a + 17\right)\cdot 19^{4} + \left(13 a^{3} + 15 a^{2} + 8 a + 3\right)\cdot 19^{5} + \left(3 a^{3} + 5 a^{2} + 17 a + 6\right)\cdot 19^{6} + \left(18 a^{3} + 12 a^{2} + 14 a + 17\right)\cdot 19^{7} + \left(2 a^{3} + 9 a^{2} + 3 a + 16\right)\cdot 19^{8} + \left(6 a^{3} + 8 a^{2} + 5 a + 16\right)\cdot 19^{9} +O(19^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 8 a^{3} + 3 a^{2} + 17 a + 7 + \left(7 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 19 + \left(4 a^{3} + 2 a^{2} + 4 a + 13\right)\cdot 19^{2} + \left(7 a^{3} + 7 a^{2} + 10 a + 12\right)\cdot 19^{3} + \left(7 a^{3} + 8 a^{2} + 2 a + 18\right)\cdot 19^{4} + \left(4 a^{2} + 3 a + 13\right)\cdot 19^{5} + \left(3 a^{3} + 6 a^{2} + 10 a + 5\right)\cdot 19^{6} + \left(11 a^{3} + a^{2} + 15 a + 13\right)\cdot 19^{7} + \left(16 a^{3} + 5 a^{2} + 11 a + 13\right)\cdot 19^{8} + \left(18 a^{3} + 11 a^{2} + 4 a + 10\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 a^{3} + 16 a + 17 + \left(17 a^{3} + 9 a^{2} + 10 a + 2\right)\cdot 19 + \left(2 a^{3} + 15 a^{2} + 9 a + 5\right)\cdot 19^{2} + \left(5 a^{3} + 17 a^{2} + 17 a\right)\cdot 19^{3} + \left(14 a^{3} + 9 a^{2} + 7 a + 3\right)\cdot 19^{4} + \left(3 a^{3} + a^{2} + 6 a + 8\right)\cdot 19^{5} + \left(11 a^{3} + 3 a^{2} + 12 a + 3\right)\cdot 19^{6} + \left(13 a^{3} + 11 a^{2} + 18 a + 16\right)\cdot 19^{7} + \left(4 a^{3} + 11 a^{2} + 5 a + 4\right)\cdot 19^{8} + \left(13 a^{2} + 15 a + 11\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 9 a^{3} + 4 a^{2} + 6 a + 3 + \left(12 a^{3} + 9 a^{2} + 14 a + 1\right)\cdot 19 + \left(10 a^{3} + 2 a^{2} + 9 a + 18\right)\cdot 19^{2} + \left(4 a^{3} + 10 a^{2} + 16 a + 1\right)\cdot 19^{3} + \left(14 a^{3} + 7 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(15 a^{3} + 4 a^{2} + 14 a + 10\right)\cdot 19^{5} + \left(15 a^{3} + 9 a^{2} + 18 a + 14\right)\cdot 19^{6} + \left(7 a^{3} + 9 a^{2} + 18 a + 9\right)\cdot 19^{7} + \left(3 a^{3} + 16 a^{2} + 3 a + 8\right)\cdot 19^{8} + \left(a^{3} + 13 a + 1\right)\cdot 19^{9} +O(19^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,6)(2,8)(3,7)(4,9)$ | $0$ |
| $36$ | $2$ | $(3,4)(5,6)(7,8)$ | $-2$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)(7,8,9)$ | $-1$ |
| $24$ | $3$ | $(1,2,9)(4,6,8)$ | $2$ |
| $48$ | $3$ | $(1,8,2)(3,5,9)(4,6,7)$ | $-1$ |
| $54$ | $4$ | $(1,4,6,9)(2,7,8,3)$ | $0$ |
| $72$ | $6$ | $(1,7,2,3,9,5)(4,6)$ | $0$ |
| $72$ | $6$ | $(1,2,9)(3,8,5,4,7,6)$ | $1$ |
| $54$ | $8$ | $(1,2,4,7,6,8,9,3)$ | $0$ |
| $54$ | $8$ | $(1,8,4,3,6,2,9,7)$ | $0$ |