Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(27556\)\(\medspace = 2^{2} \cdot 83^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.2287148.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.1328.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 19 + \left(15 a + 13\right)\cdot 41 + \left(35 a + 24\right)\cdot 41^{2} + \left(7 a + 16\right)\cdot 41^{3} + \left(25 a + 8\right)\cdot 41^{4} + \left(4 a + 9\right)\cdot 41^{5} + \left(7 a + 17\right)\cdot 41^{6} + \left(26 a + 28\right)\cdot 41^{7} + \left(4 a + 32\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 39 + \left(33 a + 13\right)\cdot 41 + \left(11 a + 40\right)\cdot 41^{2} + \left(24 a + 30\right)\cdot 41^{3} + \left(5 a + 24\right)\cdot 41^{4} + \left(8 a + 32\right)\cdot 41^{5} + 19\cdot 41^{6} + \left(25 a + 22\right)\cdot 41^{7} + \left(14 a + 37\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 17\cdot 41 + 31\cdot 41^{2} + 34\cdot 41^{3} + 41^{4} + 30\cdot 41^{5} + 33\cdot 41^{6} + 7\cdot 41^{7} + 5\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 31 + \left(7 a + 20\right)\cdot 41 + \left(29 a + 1\right)\cdot 41^{2} + \left(16 a + 10\right)\cdot 41^{3} + \left(35 a + 17\right)\cdot 41^{4} + \left(32 a + 10\right)\cdot 41^{5} + \left(40 a + 12\right)\cdot 41^{6} + \left(15 a + 15\right)\cdot 41^{7} + \left(26 a + 15\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 17\cdot 41 + 33\cdot 41^{2} + 25\cdot 41^{3} + 35\cdot 41^{4} + 41^{5} + 6\cdot 41^{6} + 31\cdot 41^{7} + 11\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a + 38 + \left(25 a + 39\right)\cdot 41 + \left(5 a + 32\right)\cdot 41^{2} + \left(33 a + 4\right)\cdot 41^{3} + \left(15 a + 35\right)\cdot 41^{4} + \left(36 a + 38\right)\cdot 41^{5} + \left(33 a + 33\right)\cdot 41^{6} + \left(14 a + 17\right)\cdot 41^{7} + \left(36 a + 20\right)\cdot 41^{8} +O(41^{9})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,3)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,5,4,3)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,3,2)$ | $0$ |