Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(465\)\(\medspace = 3 \cdot 5 \cdot 31 \) |
| Artin stem field: | Galois closure of 6.0.3243375.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.465.6t1.b.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.14415.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} - 3x^{3} + 9x^{2} - 5x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 12 + \left(4 a + 6\right)\cdot 29 + \left(7 a + 26\right)\cdot 29^{2} + \left(10 a + 3\right)\cdot 29^{3} + \left(11 a + 5\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a + 7 + \left(a + 9\right)\cdot 29 + \left(12 a + 4\right)\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + \left(23 a + 28\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 17 + \left(9 a + 8\right)\cdot 29 + \left(17 a + 16\right)\cdot 29^{2} + \left(18 a + 17\right)\cdot 29^{3} + \left(12 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 16 + \left(27 a + 22\right)\cdot 29 + \left(16 a + 4\right)\cdot 29^{2} + \left(9 a + 19\right)\cdot 29^{3} + \left(5 a + 11\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 8 + \left(19 a + 21\right)\cdot 29 + \left(11 a + 6\right)\cdot 29^{2} + \left(10 a + 6\right)\cdot 29^{3} + \left(16 a + 17\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 28 + \left(24 a + 18\right)\cdot 29 + \left(21 a + 28\right)\cdot 29^{2} + \left(18 a + 18\right)\cdot 29^{3} + \left(17 a + 22\right)\cdot 29^{4} +O(29^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-2 \zeta_{3} - 2$ | |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $2 \zeta_{3}$ | |
| $2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ | |
| $2$ | $3$ | $(4,5,6)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(4,6,5)$ | $-\zeta_{3}$ | |
| $3$ | $6$ | $(1,6,3,5,2,4)$ | $0$ | |
| $3$ | $6$ | $(1,4,2,5,3,6)$ | $0$ |