Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(2997\)\(\medspace = 3^{4} \cdot 37 \) |
| Artin number field: | Galois closure of 6.0.26946027.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.4107.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 a + 40 + \left(34 a + 4\right)\cdot 41 + \left(29 a + 36\right)\cdot 41^{2} + \left(18 a + 19\right)\cdot 41^{3} + \left(26 a + 27\right)\cdot 41^{4} + \left(11 a + 15\right)\cdot 41^{5} + \left(32 a + 8\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 5 + 8\cdot 41 + \left(16 a + 30\right)\cdot 41^{2} + \left(11 a + 19\right)\cdot 41^{3} + \left(12 a + 39\right)\cdot 41^{4} + \left(31 a + 6\right)\cdot 41^{5} + 28\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 a + 21 + \left(40 a + 31\right)\cdot 41 + \left(24 a + 36\right)\cdot 41^{2} + \left(29 a + 37\right)\cdot 41^{3} + \left(28 a + 23\right)\cdot 41^{4} + \left(9 a + 6\right)\cdot 41^{5} + \left(40 a + 40\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 10 + \left(6 a + 37\right)\cdot 41 + \left(11 a + 8\right)\cdot 41^{2} + \left(22 a + 5\right)\cdot 41^{3} + \left(14 a + 6\right)\cdot 41^{4} + \left(29 a + 24\right)\cdot 41^{5} + \left(8 a + 11\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 a + 21 + \left(6 a + 4\right)\cdot 41 + \left(27 a + 9\right)\cdot 41^{2} + \left(33 a + 24\right)\cdot 41^{3} + \left(26 a + 30\right)\cdot 41^{4} + \left(19 a + 18\right)\cdot 41^{5} + \left(9 a + 33\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 26 + \left(34 a + 36\right)\cdot 41 + \left(13 a + 1\right)\cdot 41^{2} + \left(7 a + 16\right)\cdot 41^{3} + \left(14 a + 36\right)\cdot 41^{4} + \left(21 a + 9\right)\cdot 41^{5} + \left(31 a + 1\right)\cdot 41^{6} +O(41^{7})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,5)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,3)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,6,3,4,5,2)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,2,5,4,3,6)$ | $0$ | $0$ |