Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(243\)\(\medspace = 3^{5} \) |
| Artin number field: | Galois closure of 6.0.177147.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.243.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 9 + \left(6 a + 9\right)\cdot 17 + \left(8 a + 7\right)\cdot 17^{2} + \left(16 a + 16\right)\cdot 17^{3} + \left(4 a + 6\right)\cdot 17^{4} + \left(16 a + 1\right)\cdot 17^{5} + \left(4 a + 12\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + \left(7 a + 10\right)\cdot 17 + 12\cdot 17^{2} + \left(15 a + 12\right)\cdot 17^{3} + \left(13 a + 11\right)\cdot 17^{4} + \left(a + 5\right)\cdot 17^{5} + a\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 5 + \left(9 a + 12\right)\cdot 17 + \left(16 a + 5\right)\cdot 17^{2} + \left(a + 10\right)\cdot 17^{3} + \left(3 a + 10\right)\cdot 17^{4} + \left(15 a + 10\right)\cdot 17^{5} + \left(15 a + 16\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 7 + \left(10 a + 1\right)\cdot 17 + \left(8 a + 9\right)\cdot 17^{2} + 7\cdot 17^{3} + \left(12 a + 12\right)\cdot 17^{4} + 12\cdot 17^{5} + 12 a\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 3 + 12\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + \left(15 a + 7\right)\cdot 17^{3} + \left(8 a + 16\right)\cdot 17^{4} + \left(2 a + 4\right)\cdot 17^{5} + \left(13 a + 5\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 10 + \left(16 a + 5\right)\cdot 17 + \left(7 a + 12\right)\cdot 17^{2} + \left(a + 13\right)\cdot 17^{3} + \left(8 a + 9\right)\cdot 17^{4} + \left(14 a + 15\right)\cdot 17^{5} + \left(3 a + 15\right)\cdot 17^{6} +O(17^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ | $-1$ |
| $2$ | $3$ | $(2,6,4)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,4,6)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $3$ | $6$ | $(1,4,3,2,5,6)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,6,5,2,3,4)$ | $0$ | $0$ |