Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1340\)\(\medspace = 2^{2} \cdot 5 \cdot 67 \) |
| Artin stem field: | Galois closure of 6.0.35912000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.1340.6t1.a.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.89780.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 7x^{4} - 16x^{3} + 64x^{2} - 30x + 25 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 48 a + 7 + \left(24 a + 2\right)\cdot 53 + \left(25 a + 11\right)\cdot 53^{2} + \left(6 a + 20\right)\cdot 53^{3} + \left(42 a + 36\right)\cdot 53^{4} + \left(52 a + 9\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 40 + 28 a\cdot 53 + \left(27 a + 35\right)\cdot 53^{2} + \left(46 a + 20\right)\cdot 53^{3} + \left(10 a + 39\right)\cdot 53^{4} + 19\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 30 + \left(18 a + 47\right)\cdot 53 + \left(24 a + 32\right)\cdot 53^{2} + \left(37 a + 7\right)\cdot 53^{3} + \left(23 a + 39\right)\cdot 53^{4} + \left(10 a + 16\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 46 a + 37 + \left(6 a + 4\right)\cdot 53 + \left(a + 38\right)\cdot 53^{2} + \left(22 a + 24\right)\cdot 53^{3} + \left(18 a + 27\right)\cdot 53^{4} + \left(42 a + 16\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 a + 38 + \left(34 a + 11\right)\cdot 53 + \left(28 a + 6\right)\cdot 53^{2} + \left(15 a + 27\right)\cdot 53^{3} + \left(29 a + 43\right)\cdot 53^{4} + \left(42 a + 34\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 9 + \left(46 a + 39\right)\cdot 53 + \left(51 a + 35\right)\cdot 53^{2} + \left(30 a + 5\right)\cdot 53^{3} + \left(34 a + 26\right)\cdot 53^{4} + \left(10 a + 8\right)\cdot 53^{5} +O(53^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,5,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,5)(2,3,4)$ | $-1$ |
| $3$ | $6$ | $(1,2,6,4,5,3)$ | $0$ |
| $3$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.