Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1053\)\(\medspace = 3^{4} \cdot 13 \) |
| Artin stem field: | Galois closure of 6.0.3326427.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.117.6t1.a.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.4563.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 7x^{3} + 13 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 15 + \left(19 a + 1\right)\cdot 23 + \left(12 a + 6\right)\cdot 23^{2} + \left(15 a + 1\right)\cdot 23^{3} + \left(11 a + 5\right)\cdot 23^{4} + \left(17 a + 17\right)\cdot 23^{5} + \left(6 a + 21\right)\cdot 23^{6} + 8\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 3 + \left(3 a + 1\right)\cdot 23 + \left(10 a + 12\right)\cdot 23^{2} + \left(7 a + 19\right)\cdot 23^{3} + \left(11 a + 12\right)\cdot 23^{4} + \left(5 a + 17\right)\cdot 23^{5} + \left(16 a + 17\right)\cdot 23^{6} + \left(22 a + 2\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 18 + \left(5 a + 4\right)\cdot 23 + \left(18 a + 11\right)\cdot 23^{2} + \left(19 a + 7\right)\cdot 23^{3} + 16 a\cdot 23^{4} + \left(21 a + 21\right)\cdot 23^{5} + \left(7 a + 20\right)\cdot 23^{6} + \left(a + 22\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 16 + \left(2 a + 21\right)\cdot 23 + \left(8 a + 14\right)\cdot 23^{2} + \left(12 a + 20\right)\cdot 23^{3} + \left(5 a + 18\right)\cdot 23^{4} + \left(16 a + 3\right)\cdot 23^{5} + \left(14 a + 13\right)\cdot 23^{6} + \left(a + 2\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 13 + \left(20 a + 16\right)\cdot 23 + \left(14 a + 5\right)\cdot 23^{2} + \left(10 a + 14\right)\cdot 23^{3} + \left(17 a + 17\right)\cdot 23^{4} + \left(6 a + 7\right)\cdot 23^{5} + \left(8 a + 3\right)\cdot 23^{6} + \left(21 a + 14\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 4 + 17 a\cdot 23 + \left(4 a + 19\right)\cdot 23^{2} + \left(3 a + 5\right)\cdot 23^{3} + \left(6 a + 14\right)\cdot 23^{4} + \left(a + 1\right)\cdot 23^{5} + \left(15 a + 15\right)\cdot 23^{6} + \left(21 a + 17\right)\cdot 23^{7} +O(23^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-2 \zeta_{3} - 2$ | |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $2 \zeta_{3}$ | |
| $2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ | |
| $3$ | $6$ | $(1,2,3,6,5,4)$ | $0$ | |
| $3$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |