Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(989\)\(\medspace = 23 \cdot 43 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.22747.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.42527.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 19 + 33\cdot 59 + \left(43 a + 57\right)\cdot 59^{2} + \left(58 a + 39\right)\cdot 59^{3} + \left(8 a + 25\right)\cdot 59^{4} + \left(30 a + 4\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 a + 47 + \left(58 a + 5\right)\cdot 59 + \left(15 a + 41\right)\cdot 59^{2} + 55\cdot 59^{3} + \left(50 a + 34\right)\cdot 59^{4} + \left(28 a + 25\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 32 + \left(17 a + 49\right)\cdot 59 + \left(7 a + 18\right)\cdot 59^{2} + \left(9 a + 49\right)\cdot 59^{3} + \left(56 a + 8\right)\cdot 59^{4} + \left(39 a + 15\right)\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 20\cdot 59^{2} + 4\cdot 59^{3} + 39\cdot 59^{4} + 51\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 31\cdot 59^{2} + 35\cdot 59^{3} + 12\cdot 59^{4} + 22\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 12 + \left(41 a + 28\right)\cdot 59 + \left(51 a + 8\right)\cdot 59^{2} + \left(49 a + 51\right)\cdot 59^{3} + \left(2 a + 55\right)\cdot 59^{4} + \left(19 a + 57\right)\cdot 59^{5} +O(59^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,2,3)$ | $1$ |
| $6$ | $4$ | $(1,5,2,4)(3,6)$ | $-1$ |
| $8$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |