Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Artin number field: | Galois closure of 6.0.3136000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1960.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 20 + \left(2 a + 7\right)\cdot 29 + \left(3 a + 10\right)\cdot 29^{2} + \left(3 a + 22\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 8 + \left(4 a + 6\right)\cdot 29 + \left(16 a + 28\right)\cdot 29^{2} + \left(10 a + 14\right)\cdot 29^{3} + \left(a + 9\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + 2 + \left(21 a + 15\right)\cdot 29 + \left(9 a + 19\right)\cdot 29^{2} + \left(15 a + 20\right)\cdot 29^{3} + \left(20 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 6 + \left(7 a + 13\right)\cdot 29 + \left(19 a + 17\right)\cdot 29^{2} + 13 a\cdot 29^{3} + \left(8 a + 8\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 4 + \left(26 a + 1\right)\cdot 29 + \left(25 a + 23\right)\cdot 29^{2} + \left(25 a + 5\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 20 + \left(24 a + 14\right)\cdot 29 + \left(12 a + 17\right)\cdot 29^{2} + \left(18 a + 22\right)\cdot 29^{3} + \left(27 a + 5\right)\cdot 29^{4} +O(29^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,2)(4,5,6)$ | $-1$ | $-1$ |
| $2$ | $3$ | $(4,6,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(4,5,6)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $3$ | $6$ | $(1,5,2,6,3,4)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,4,3,6,2,5)$ | $0$ | $0$ |