Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(5585081\)\(\medspace = 29^{3} \cdot 229 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.1278983549.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.0.229.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 37\cdot 53 + 49\cdot 53^{2} + 32\cdot 53^{3} + 39\cdot 53^{4} + 6\cdot 53^{5} + 34\cdot 53^{6} + 7\cdot 53^{7} + 7\cdot 53^{8} + 12\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 16 + \left(41 a + 52\right)\cdot 53 + \left(7 a + 18\right)\cdot 53^{2} + \left(38 a + 11\right)\cdot 53^{3} + \left(16 a + 30\right)\cdot 53^{4} + \left(26 a + 34\right)\cdot 53^{5} + \left(30 a + 34\right)\cdot 53^{6} + \left(7 a + 17\right)\cdot 53^{7} + \left(13 a + 16\right)\cdot 53^{8} + \left(4 a + 18\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a + 7 + \left(11 a + 47\right)\cdot 53 + \left(45 a + 8\right)\cdot 53^{2} + \left(14 a + 50\right)\cdot 53^{3} + \left(36 a + 5\right)\cdot 53^{4} + \left(26 a + 17\right)\cdot 53^{5} + \left(22 a + 24\right)\cdot 53^{6} + \left(45 a + 17\right)\cdot 53^{7} + \left(39 a + 8\right)\cdot 53^{8} + \left(48 a + 22\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 6\cdot 53 + 20\cdot 53^{2} + 28\cdot 53^{3} + 22\cdot 53^{4} + 45\cdot 53^{5} + 35\cdot 53^{6} + 5\cdot 53^{7} + 46\cdot 53^{8} + 44\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 43 + \left(33 a + 48\right)\cdot 53 + \left(50 a + 51\right)\cdot 53^{2} + \left(44 a + 32\right)\cdot 53^{3} + \left(24 a + 29\right)\cdot 53^{4} + \left(50 a + 18\right)\cdot 53^{5} + \left(50 a + 44\right)\cdot 53^{6} + \left(31 a + 16\right)\cdot 53^{7} + \left(5 a + 45\right)\cdot 53^{8} + \left(24 a + 11\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 a + 51 + \left(19 a + 19\right)\cdot 53 + \left(2 a + 9\right)\cdot 53^{2} + \left(8 a + 3\right)\cdot 53^{3} + \left(28 a + 31\right)\cdot 53^{4} + \left(2 a + 36\right)\cdot 53^{5} + \left(2 a + 38\right)\cdot 53^{6} + \left(21 a + 40\right)\cdot 53^{7} + \left(47 a + 35\right)\cdot 53^{8} + \left(28 a + 49\right)\cdot 53^{9} +O(53^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $6$ | $2$ | $(1,5)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,2,4,3)$ | $1$ |
| $6$ | $4$ | $(1,2,4,3)(5,6)$ | $-1$ |
| $8$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |