Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(37423\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.37423.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Projective image: | $S_6$ |
| Projective field: | Galois closure of 6.0.37423.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 23 + \left(17 a + 39\right)\cdot 41 + \left(5 a + 6\right)\cdot 41^{2} + \left(36 a + 10\right)\cdot 41^{3} + \left(17 a + 37\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 6 + \left(13 a + 10\right)\cdot 41 + \left(12 a + 4\right)\cdot 41^{2} + \left(37 a + 21\right)\cdot 41^{3} + \left(5 a + 9\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 7 + \left(23 a + 29\right)\cdot 41 + \left(35 a + 5\right)\cdot 41^{2} + \left(4 a + 31\right)\cdot 41^{3} + \left(23 a + 13\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 20\cdot 41 + 2\cdot 41^{2} + 30\cdot 41^{3} + 10\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 + 20\cdot 41 + 34\cdot 41^{2} + 32\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 24 + \left(27 a + 2\right)\cdot 41 + \left(28 a + 28\right)\cdot 41^{2} + \left(3 a + 38\right)\cdot 41^{3} + \left(35 a + 30\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |