Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1053\)\(\medspace = 3^{4} \cdot 13 \) |
| Artin stem field: | Galois closure of 6.0.43243551.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.117.6t1.h.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.351.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} - 3x^{3} + 9x^{2} - 9x + 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 15 + \left(11 a + 15\right)\cdot 17 + \left(6 a + 5\right)\cdot 17^{2} + \left(6 a + 10\right)\cdot 17^{3} + \left(5 a + 9\right)\cdot 17^{4} + \left(6 a + 9\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 5 + \left(11 a + 3\right)\cdot 17 + \left(12 a + 1\right)\cdot 17^{2} + \left(11 a + 6\right)\cdot 17^{3} + \left(2 a + 2\right)\cdot 17^{4} + \left(14 a + 7\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + \left(5 a + 6\right)\cdot 17 + \left(2 a + 13\right)\cdot 17^{2} + \left(a + 1\right)\cdot 17^{3} + \left(8 a + 15\right)\cdot 17^{4} + \left(3 a + 4\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 14 + \left(11 a + 14\right)\cdot 17 + \left(14 a + 9\right)\cdot 17^{2} + 15 a\cdot 17^{3} + \left(8 a + 5\right)\cdot 17^{4} + 13 a\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 11 + \left(5 a + 14\right)\cdot 17 + 10 a\cdot 17^{2} + \left(10 a + 10\right)\cdot 17^{3} + \left(11 a + 8\right)\cdot 17^{4} + \left(10 a + 10\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 6 + \left(5 a + 13\right)\cdot 17 + \left(4 a + 2\right)\cdot 17^{2} + \left(5 a + 5\right)\cdot 17^{3} + \left(14 a + 10\right)\cdot 17^{4} + \left(2 a + 1\right)\cdot 17^{5} +O(17^{6})\)
|
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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,2,4)(3,5,6)$ | $2 \zeta_{3}$ | |
| $1$ | $3$ | $(1,4,2)(3,6,5)$ | $-2 \zeta_{3} - 2$ | |
| $2$ | $3$ | $(1,2,4)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,4,2)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ | |
| $3$ | $6$ | $(1,5,4,3,2,6)$ | $0$ | |
| $3$ | $6$ | $(1,6,2,3,4,5)$ | $0$ |