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Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(364\)\(\medspace = 2^{2} \cdot 7 \cdot 13 \) |
| Artin number field: | Galois closure of 9.3.16542390592.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.2548.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 4 + \left(5 a^{2} + 5 a + 9\right)\cdot 11 + \left(5 a^{2} + a + 4\right)\cdot 11^{2} + \left(4 a^{2} + 9 a + 3\right)\cdot 11^{3} + \left(8 a^{2} + 9 a + 6\right)\cdot 11^{4} + \left(9 a^{2} + 4 a + 2\right)\cdot 11^{5} + \left(7 a^{2} + 8 a + 2\right)\cdot 11^{6} + \left(3 a^{2} + 5 a + 6\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a^{2} + 4 a + 3 + \left(2 a^{2} + a + 7\right)\cdot 11 + \left(4 a^{2} + a + 2\right)\cdot 11^{2} + \left(7 a^{2} + 8\right)\cdot 11^{3} + \left(9 a^{2} + a + 5\right)\cdot 11^{4} + \left(7 a + 5\right)\cdot 11^{5} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{6} + \left(8 a + 5\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( a^{2} + 3 a + 4 + \left(a^{2} + 10 a\right)\cdot 11 + \left(a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(7 a^{2} + 6 a + 6\right)\cdot 11^{3} + \left(8 a^{2} + a + 6\right)\cdot 11^{4} + \left(3 a^{2} + 6 a + 5\right)\cdot 11^{5} + \left(3 a^{2} + 4 a + 3\right)\cdot 11^{6} + \left(2 a^{2} + 5 a + 4\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 5 + \left(10 a^{2} + 8 a + 6\right)\cdot 11 + \left(3 a^{2} + 7 a + 9\right)\cdot 11^{2} + \left(8 a^{2} + 5 a + 5\right)\cdot 11^{3} + \left(2 a^{2} + 7 a + 7\right)\cdot 11^{4} + \left(2 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(9 a^{2} + 8 a + 9\right)\cdot 11^{6} + \left(4 a^{2} + 7\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a^{2} + 10 a + 8 + \left(3 a^{2} + 2 a + 5\right)\cdot 11 + \left(2 a^{2} + 8\right)\cdot 11^{2} + \left(4 a^{2} + 5 a + 9\right)\cdot 11^{3} + \left(7 a^{2} + 10 a + 10\right)\cdot 11^{4} + \left(a^{2} + 3 a\right)\cdot 11^{5} + \left(9 a^{2} + 9 a + 5\right)\cdot 11^{6} + \left(5 a^{2} + 2 a + 1\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a^{2} + 7 a + 5 + \left(8 a^{2} + a + 7\right)\cdot 11 + \left(2 a^{2} + 2 a\right)\cdot 11^{2} + \left(6 a^{2} + 5 a + 3\right)\cdot 11^{3} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{4} + \left(7 a^{2} + 10 a + 3\right)\cdot 11^{5} + \left(10 a^{2} + 9 a\right)\cdot 11^{6} + \left(5 a^{2} + a + 2\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 4 a^{2} + 2 + \left(6 a^{2} + 9 a + 6\right)\cdot 11 + \left(a^{2} + a + 7\right)\cdot 11^{2} + \left(9 a^{2} + 7 a + 1\right)\cdot 11^{3} + \left(10 a^{2} + 4 a + 8\right)\cdot 11^{4} + \left(9 a^{2} + a + 4\right)\cdot 11^{5} + \left(4 a^{2} + 5 a + 10\right)\cdot 11^{6} + \left(2 a^{2} + 4 a + 7\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{2} + 8 a + \left(4 a^{2} + 6 a + 9\right)\cdot 11 + \left(4 a^{2} + 9 a + 10\right)\cdot 11^{2} + \left(10 a^{2} + 5 a + 3\right)\cdot 11^{3} + \left(4 a^{2} + 10 a + 5\right)\cdot 11^{4} + \left(8 a^{2} + 10 a + 4\right)\cdot 11^{5} + \left(10 a^{2} + 8 a + 2\right)\cdot 11^{6} + \left(4 a^{2} + 10 a + 4\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 9 }$ | $=$ |
\( 4 a^{2} + a + 2 + \left(a^{2} + 10 a + 3\right)\cdot 11 + \left(7 a^{2} + 8 a\right)\cdot 11^{2} + \left(8 a^{2} + 9 a + 1\right)\cdot 11^{3} + \left(3 a^{2} + 6 a + 6\right)\cdot 11^{4} + \left(10 a^{2} + 5 a + 1\right)\cdot 11^{5} + \left(7 a^{2} + 7 a + 7\right)\cdot 11^{6} + \left(2 a^{2} + 3 a + 4\right)\cdot 11^{7} +O(11^{8})\)
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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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,9)(3,4)(6,7)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)(2,3,7)(4,8,9)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,5)(2,7,3)(4,9,8)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,2,9)(3,4,5)(6,7,8)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,3,7,9,4,6)(2,5,8)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,6,4,9,7,3)(2,8,5)$ | $0$ | $0$ |