Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(2541\)\(\medspace = 3 \cdot 7 \cdot 11^{2} \) |
| Artin stem field: | Galois closure of 6.0.19370043.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.21.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.17787.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 4x^{4} + 8x^{3} - 17x^{2} + 7x + 31 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 10 + \left(12 a + 30\right)\cdot 41 + \left(2 a + 39\right)\cdot 41^{2} + \left(14 a + 24\right)\cdot 41^{3} + \left(17 a + 37\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 19 + \left(14 a + 10\right)\cdot 41 + \left(19 a + 38\right)\cdot 41^{2} + \left(33 a + 35\right)\cdot 41^{3} + \left(a + 21\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a + 28 + \left(28 a + 19\right)\cdot 41 + \left(38 a + 34\right)\cdot 41^{2} + \left(26 a + 23\right)\cdot 41^{3} + \left(23 a + 34\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 11 + \left(26 a + 1\right)\cdot 41 + 21 a\cdot 41^{2} + \left(7 a + 35\right)\cdot 41^{3} + \left(39 a + 34\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 34 + \left(23 a + 27\right)\cdot 41 + \left(15 a + 34\right)\cdot 41^{2} + \left(12 a + 31\right)\cdot 41^{3} + \left(11 a + 6\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 24 + \left(17 a + 33\right)\cdot 41 + \left(25 a + 16\right)\cdot 41^{2} + \left(28 a + 12\right)\cdot 41^{3} + \left(29 a + 28\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,4,5)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ |
| $3$ | $6$ | $(1,6,5,2,4,3)$ | $0$ |
| $3$ | $6$ | $(1,3,4,2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.