Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1352\)\(\medspace = 2^{3} \cdot 13^{2} \) |
| Artin stem field: | Galois closure of 6.0.190102016.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.104.6t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.104.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 7x^{4} + 6x^{3} + 44x^{2} + 60x + 27 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 52 a + 3 + \left(a + 12\right)\cdot 53 + \left(34 a + 30\right)\cdot 53^{2} + \left(8 a + 50\right)\cdot 53^{3} + \left(39 a + 24\right)\cdot 53^{4} + \left(41 a + 51\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 24 + \left(12 a + 23\right)\cdot 53 + \left(41 a + 11\right)\cdot 53^{2} + \left(51 a + 31\right)\cdot 53^{3} + \left(48 a + 32\right)\cdot 53^{4} + \left(47 a + 19\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 28 + \left(10 a + 34\right)\cdot 53 + \left(7 a + 17\right)\cdot 53^{2} + \left(43 a + 17\right)\cdot 53^{3} + \left(9 a + 10\right)\cdot 53^{4} + \left(6 a + 51\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a + 31 + \left(42 a + 8\right)\cdot 53 + \left(45 a + 36\right)\cdot 53^{2} + \left(9 a + 23\right)\cdot 53^{3} + \left(43 a + 6\right)\cdot 53^{4} + \left(46 a + 13\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 52 + \left(51 a + 20\right)\cdot 53 + \left(18 a + 5\right)\cdot 53^{2} + \left(44 a + 51\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} + \left(11 a + 20\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a + 23 + \left(40 a + 6\right)\cdot 53 + \left(11 a + 5\right)\cdot 53^{2} + \left(a + 38\right)\cdot 53^{3} + \left(4 a + 17\right)\cdot 53^{4} + \left(5 a + 3\right)\cdot 53^{5} +O(53^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,3,6)(2,4,5)$ | $-1$ |
| $2$ | $3$ | $(2,4,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,5,4)$ | $\zeta_{3} + 1$ |
| $3$ | $6$ | $(1,5,3,4,6,2)$ | $0$ |
| $3$ | $6$ | $(1,2,6,4,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.