Basic invariants
| Dimension: | $8$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(320557799440384\)\(\medspace = 2^{18} \cdot 11^{4} \cdot 17^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.17331195904.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T177 |
| Parity: | even |
| Projective image: | $C_3^3:S_4$ |
| Projective field: | Galois closure of 9.1.17331195904.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{3} + 2x + 18 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a^{2} + a + 4 + \left(16 a^{2} + 16 a + 14\right)\cdot 23 + \left(8 a^{2} + 12 a + 11\right)\cdot 23^{2} + \left(9 a^{2} + a + 22\right)\cdot 23^{3} + \left(18 a + 22\right)\cdot 23^{4} + \left(5 a^{2} + 4 a + 17\right)\cdot 23^{5} + \left(11 a^{2} + 7 a + 16\right)\cdot 23^{6} + \left(5 a^{2} + 22 a + 10\right)\cdot 23^{7} + \left(2 a^{2} + a + 14\right)\cdot 23^{8} + \left(18 a^{2} + 5 a + 15\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a^{2} + 8 + \left(9 a^{2} + 14 a + 15\right)\cdot 23 + \left(11 a^{2} + 8 a + 3\right)\cdot 23^{2} + \left(12 a + 5\right)\cdot 23^{3} + \left(22 a^{2} + 9 a + 14\right)\cdot 23^{4} + \left(4 a^{2} + 16 a + 13\right)\cdot 23^{5} + \left(14 a^{2} + 6 a + 5\right)\cdot 23^{6} + \left(16 a^{2} + 11 a + 14\right)\cdot 23^{7} + \left(11 a^{2} + 14 a + 14\right)\cdot 23^{8} + \left(10 a + 6\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a^{2} + 6 a + 11 + \left(5 a^{2} + 18 a + 5\right)\cdot 23 + \left(11 a^{2} + 13 a + 11\right)\cdot 23^{2} + \left(5 a^{2} + 5 a\right)\cdot 23^{3} + \left(7 a^{2} + 11 a + 10\right)\cdot 23^{4} + \left(5 a^{2} + 13 a + 19\right)\cdot 23^{5} + \left(13 a^{2} + 22 a + 13\right)\cdot 23^{6} + \left(9 a^{2} + 15 a + 9\right)\cdot 23^{7} + \left(10 a^{2} + 6 a + 3\right)\cdot 23^{8} + \left(8 a^{2} + 7 a + 6\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a^{2} + 21 a + 1 + \left(17 a^{2} + 16 a + 5\right)\cdot 23 + \left(3 a^{2} + 12 a + 1\right)\cdot 23^{2} + \left(20 a + 1\right)\cdot 23^{3} + \left(19 a^{2} + 21 a + 18\right)\cdot 23^{4} + \left(14 a^{2} + 21 a + 16\right)\cdot 23^{5} + \left(5 a^{2} + 3 a + 3\right)\cdot 23^{6} + \left(11 a^{2} + 11 a + 4\right)\cdot 23^{7} + \left(14 a^{2} + 6 a + 1\right)\cdot 23^{8} + \left(22 a^{2} + 12 a + 2\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a^{2} + a + 18 + \left(12 a^{2} + 13 a + 3\right)\cdot 23 + \left(10 a^{2} + 20 a + 10\right)\cdot 23^{2} + \left(13 a^{2} + 22\right)\cdot 23^{3} + \left(3 a^{2} + 6 a + 4\right)\cdot 23^{4} + \left(3 a^{2} + 19 a + 11\right)\cdot 23^{5} + \left(6 a^{2} + 11 a + 2\right)\cdot 23^{6} + \left(6 a^{2} + 12 a + 8\right)\cdot 23^{7} + \left(6 a^{2} + 14 a + 7\right)\cdot 23^{8} + \left(5 a^{2} + 5 a + 5\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a^{2} + 18 a + 2 + \left(16 a^{2} + 8 a + 7\right)\cdot 23 + \left(10 a^{2} + 15 a + 14\right)\cdot 23^{2} + \left(8 a^{2} + 7 a + 13\right)\cdot 23^{3} + \left(18 a^{2} + 4 a + 8\right)\cdot 23^{4} + \left(2 a^{2} + 22 a + 7\right)\cdot 23^{5} + \left(7 a^{2} + 18 a + 11\right)\cdot 23^{6} + \left(13 a^{2} + 7 a + 13\right)\cdot 23^{7} + \left(7 a^{2} + 22 a + 21\right)\cdot 23^{8} + \left(20 a^{2} + 8 a + 10\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 19 a + 6 + \left(10 a + 5\right)\cdot 23 + \left(8 a^{2} + 19 a + 22\right)\cdot 23^{2} + \left(17 a^{2} + 19 a\right)\cdot 23^{3} + \left(19 a^{2} + 12 a + 19\right)\cdot 23^{4} + \left(2 a^{2} + 10 a\right)\cdot 23^{5} + \left(4 a^{2} + 19 a + 17\right)\cdot 23^{6} + \left(2 a^{2} + 18 a + 22\right)\cdot 23^{7} + \left(21 a^{2} + 9 a + 9\right)\cdot 23^{8} + \left(14 a^{2} + 3 a + 22\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 10 a^{2} + 22 a + 10 + \left(18 a + 10\right)\cdot 23 + \left(a^{2} + 16 a + 20\right)\cdot 23^{2} + \left(9 a^{2} + 9 a + 8\right)\cdot 23^{3} + \left(20 a^{2} + 7 a + 4\right)\cdot 23^{4} + \left(14 a^{2} + 10 a + 19\right)\cdot 23^{5} + \left(2 a^{2} + 4 a + 20\right)\cdot 23^{6} + \left(22 a + 22\right)\cdot 23^{7} + \left(5 a^{2} + 16 a + 20\right)\cdot 23^{8} + \left(17 a^{2} + 6 a + 5\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 3 a^{2} + 4 a + 9 + \left(13 a^{2} + 21 a + 2\right)\cdot 23 + \left(3 a^{2} + 17 a + 20\right)\cdot 23^{2} + \left(5 a^{2} + 13 a + 16\right)\cdot 23^{3} + \left(4 a^{2} + 12\right)\cdot 23^{4} + \left(15 a^{2} + 19 a + 8\right)\cdot 23^{5} + \left(4 a^{2} + 19 a\right)\cdot 23^{6} + \left(4 a^{2} + 15 a + 9\right)\cdot 23^{7} + \left(13 a^{2} + 21 a + 21\right)\cdot 23^{8} + \left(7 a^{2} + 8 a + 16\right)\cdot 23^{9} +O(23^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $27$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,3)(6,7)(8,9)$ | $0$ |
| $6$ | $3$ | $(3,6,8)$ | $-4$ |
| $8$ | $3$ | $(1,4,5)(2,7,9)(3,6,8)$ | $-1$ |
| $12$ | $3$ | $(1,4,5)(3,6,8)$ | $2$ |
| $72$ | $3$ | $(1,2,3)(4,7,6)(5,9,8)$ | $-1$ |
| $54$ | $4$ | $(2,3,7,6)(8,9)$ | $0$ |
| $54$ | $6$ | $(1,5,4)(2,7)(3,6)$ | $0$ |
| $108$ | $6$ | $(1,4)(2,3,9,8,7,6)$ | $0$ |
| $72$ | $9$ | $(1,2,3,4,7,6,5,9,8)$ | $-1$ |
| $72$ | $9$ | $(1,2,3,5,9,8,4,7,6)$ | $2$ |
| $54$ | $12$ | $(1,4,5)(2,3,7,6)(8,9)$ | $0$ |
| $54$ | $12$ | $(1,5,4)(2,3,7,6)(8,9)$ | $0$ |