Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Artin stem field: | Galois closure of 9.3.308915776.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.52.6t1.b.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.676.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 2x^{7} - x^{6} + 5x^{5} + x^{4} - 5x^{3} + 2x^{2} + 2x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 8 + \left(9 a^{2} + 2 a + 2\right)\cdot 17 + \left(8 a^{2} + 12 a + 4\right)\cdot 17^{2} + \left(12 a^{2} + 15 a + 12\right)\cdot 17^{3} + \left(6 a^{2} + 14 a + 14\right)\cdot 17^{4} + \left(16 a^{2} + 13 a + 9\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a^{2} + 8 a + 9 + \left(9 a^{2} + 7 a + 8\right)\cdot 17 + \left(4 a^{2} + 8 a + 1\right)\cdot 17^{2} + \left(3 a^{2} + 9 a + 6\right)\cdot 17^{3} + \left(2 a^{2} + 16 a\right)\cdot 17^{4} + \left(15 a^{2} + 15 a + 9\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 4 + \left(13 a^{2} + a + 1\right)\cdot 17 + \left(12 a^{2} + 13 a\right)\cdot 17^{2} + \left(16 a^{2} + 13 a\right)\cdot 17^{3} + \left(10 a^{2} + 12 a + 13\right)\cdot 17^{4} + \left(4 a^{2} + 10 a + 16\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{2} + 4 a + 13 + \left(13 a^{2} + 12 a + 12\right)\cdot 17 + \left(16 a^{2} + 10 a + 2\right)\cdot 17^{2} + \left(2 a^{2} + a + 2\right)\cdot 17^{3} + \left(a^{2} + 4 a + 12\right)\cdot 17^{4} + \left(13 a^{2} + 9 a + 16\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a^{2} + 2 a + 16 + \left(4 a^{2} + 4 a + 2\right)\cdot 17 + \left(5 a^{2} + 10 a + 2\right)\cdot 17^{2} + \left(8 a^{2} + 2 a + 7\right)\cdot 17^{3} + \left(a + 1\right)\cdot 17^{4} + \left(2 a^{2} + 6 a\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a^{2} + 3 a + 7 + \left(15 a^{2} + 7 a + 12\right)\cdot 17 + \left(3 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(a^{2} + 8 a + 10\right)\cdot 17^{3} + \left(8 a^{2} + 2 a + 15\right)\cdot 17^{4} + \left(2 a^{2} + 4 a + 11\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 4 a + 11 + \left(6 a^{2} + 3\right)\cdot 17 + \left(9 a^{2} + 2 a + 16\right)\cdot 17^{2} + \left(7 a^{2} + 5 a\right)\cdot 17^{3} + \left(15 a^{2} + 11 a\right)\cdot 17^{4} + \left(14 a + 5\right)\cdot 17^{5} +O(17^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 12 a^{2} + 10 a + 12 + \left(7 a^{2} + 3 a + 14\right)\cdot 17 + \left(4 a^{2} + 10 a + 5\right)\cdot 17^{2} + \left(14 a^{2} + a + 15\right)\cdot 17^{3} + \left(4 a^{2} + 8\right)\cdot 17^{4} + \left(16 a^{2} + 14 a + 7\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 9 }$ | $=$ |
\( a^{2} + 11 a + 6 + \left(6 a^{2} + 12 a + 9\right)\cdot 17 + \left(2 a^{2} + 4 a + 11\right)\cdot 17^{2} + \left(a^{2} + 9 a + 13\right)\cdot 17^{3} + \left(a^{2} + 4 a + 1\right)\cdot 17^{4} + \left(14 a^{2} + 13 a + 8\right)\cdot 17^{5} +O(17^{6})\)
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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 value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(6,9)(7,8)$ | $0$ |
| $1$ | $3$ | $(1,9,8)(2,5,3)(4,6,7)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,8,9)(2,3,5)(4,7,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,2,6)(3,4,8)(5,7,9)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,2)(3,8,4)(5,9,7)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,4)(2,7,8)(3,6,9)$ | $-1$ |
| $3$ | $6$ | $(1,7,9,4,8,6)(2,3,5)$ | $0$ |
| $3$ | $6$ | $(1,6,8,4,9,7)(2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.