Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Artin number field: | Galois closure of 6.0.73008.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.2028.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 21 + \left(31 a + 30\right)\cdot 47 + \left(4 a + 7\right)\cdot 47^{2} + \left(17 a + 12\right)\cdot 47^{3} + \left(33 a + 32\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 12 + \left(6 a + 36\right)\cdot 47 + \left(22 a + 1\right)\cdot 47^{2} + \left(18 a + 42\right)\cdot 47^{3} + \left(36 a + 20\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a + 22 + \left(40 a + 43\right)\cdot 47 + \left(24 a + 39\right)\cdot 47^{2} + \left(28 a + 9\right)\cdot 47^{3} + \left(10 a + 28\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 17 + \left(41 a + 6\right)\cdot 47 + \left(38 a + 11\right)\cdot 47^{2} + \left(27 a + 9\right)\cdot 47^{3} + \left(23 a + 26\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a + 41 + \left(5 a + 29\right)\cdot 47 + 8 a\cdot 47^{2} + \left(19 a + 26\right)\cdot 47^{3} + \left(23 a + 45\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 43 a + 29 + \left(15 a + 41\right)\cdot 47 + \left(42 a + 32\right)\cdot 47^{2} + \left(29 a + 41\right)\cdot 47^{3} + \left(13 a + 34\right)\cdot 47^{4} +O(47^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,5,2)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,2,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,2,5)(3,6,4)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,3,5,6,2,4)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,4,2,6,5,3)$ | $0$ | $0$ |