Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(804\)\(\medspace = 2^{2} \cdot 3 \cdot 67 \) |
| Artin number field: | Galois closure of 6.0.1939248.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.53868.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 40 a + 43 + \left(28 a + 33\right)\cdot 53 + \left(39 a + 10\right)\cdot 53^{2} + \left(7 a + 50\right)\cdot 53^{3} + \left(18 a + 38\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 46 + \left(40 a + 14\right)\cdot 53 + \left(48 a + 49\right)\cdot 53^{2} + \left(35 a + 52\right)\cdot 53^{3} + \left(33 a + 48\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 a + 12 + \left(26 a + 46\right)\cdot 53 + \left(42 a + 17\right)\cdot 53^{2} + \left(38 a + 9\right)\cdot 53^{3} + \left(34 a + 45\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 44 + \left(24 a + 2\right)\cdot 53 + \left(13 a + 34\right)\cdot 53^{2} + \left(45 a + 41\right)\cdot 53^{3} + \left(34 a + 50\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 50 + \left(26 a + 10\right)\cdot 53 + \left(10 a + 2\right)\cdot 53^{2} + \left(14 a + 16\right)\cdot 53^{3} + \left(18 a + 39\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 20 + \left(12 a + 50\right)\cdot 53 + \left(4 a + 44\right)\cdot 53^{2} + \left(17 a + 41\right)\cdot 53^{3} + \left(19 a + 41\right)\cdot 53^{4} +O(53^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(3,4,6)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(3,6,4)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,2,5)(3,6,4)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,6,2,3,5,4)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,4,5,3,2,6)$ | $0$ | $0$ |