Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(25747\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.25747.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Projective image: | $S_6$ |
| Projective field: | Galois closure of 6.0.25747.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 3 a + 14 + \left(20 a + 3\right)\cdot 41 + \left(13 a + 39\right)\cdot 41^{2} + \left(38 a + 40\right)\cdot 41^{3} + \left(16 a + 22\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 27\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 32\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a + \left(2 a + 30\right)\cdot 41 + \left(38 a + 35\right)\cdot 41^{2} + \left(40 a + 35\right)\cdot 41^{3} + \left(35 a + 25\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 23 + \left(38 a + 3\right)\cdot 41 + \left(2 a + 24\right)\cdot 41^{2} + 38\cdot 41^{3} + \left(5 a + 10\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 39\cdot 41 + 16\cdot 41^{2} + 25\cdot 41^{3} + 36\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 38 a + 23 + \left(20 a + 19\right)\cdot 41 + \left(27 a + 18\right)\cdot 41^{2} + \left(2 a + 19\right)\cdot 41^{3} + \left(24 a + 35\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |