Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
| Artin number field: | Galois closure of 6.0.292032000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.20280.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 45 a + 16 + \left(30 a + 3\right)\cdot 53 + 43 a\cdot 53^{2} + \left(50 a + 6\right)\cdot 53^{3} + \left(22 a + 1\right)\cdot 53^{4} + \left(9 a + 2\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 a + 37 + \left(14 a + 9\right)\cdot 53 + \left(51 a + 44\right)\cdot 53^{2} + \left(52 a + 8\right)\cdot 53^{3} + \left(36 a + 14\right)\cdot 53^{4} + \left(29 a + 45\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 a + 1 + \left(7 a + 40\right)\cdot 53 + \left(11 a + 8\right)\cdot 53^{2} + \left(2 a + 38\right)\cdot 53^{3} + \left(46 a + 37\right)\cdot 53^{4} + \left(13 a + 5\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 37 + \left(22 a + 28\right)\cdot 53 + \left(9 a + 37\right)\cdot 53^{2} + \left(2 a + 6\right)\cdot 53^{3} + \left(30 a + 42\right)\cdot 53^{4} + \left(43 a + 16\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 a + 23 + \left(45 a + 38\right)\cdot 53 + \left(41 a + 45\right)\cdot 53^{2} + \left(50 a + 35\right)\cdot 53^{3} + \left(6 a + 7\right)\cdot 53^{4} + \left(39 a + 15\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 47 + \left(38 a + 38\right)\cdot 53 + \left(a + 22\right)\cdot 53^{2} + 10\cdot 53^{3} + \left(16 a + 3\right)\cdot 53^{4} + \left(23 a + 21\right)\cdot 53^{5} +O(53^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ | $-1$ |
| $2$ | $3$ | $(4,6,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(4,5,6)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $3$ | $6$ | $(1,6,3,4,2,5)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,5,2,4,3,6)$ | $0$ | $0$ |