Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(46651\)\(\medspace = 11 \cdot 4241 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.46651.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Projective image: | $S_6$ |
| Projective field: | Galois closure of 6.0.46651.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 15\cdot 61 + 30\cdot 61^{2} + 46\cdot 61^{3} + 31\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 29 + \left(19 a + 58\right)\cdot 61 + \left(7 a + 46\right)\cdot 61^{2} + \left(41 a + 14\right)\cdot 61^{3} + \left(41 a + 23\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 25 + \left(26 a + 39\right)\cdot 61 + \left(5 a + 57\right)\cdot 61^{2} + \left(29 a + 34\right)\cdot 61^{3} + \left(34 a + 22\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 a + 48 + \left(41 a + 58\right)\cdot 61 + \left(53 a + 34\right)\cdot 61^{2} + \left(19 a + 48\right)\cdot 61^{3} + \left(19 a + 23\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 a + 44 + \left(34 a + 46\right)\cdot 61 + \left(55 a + 36\right)\cdot 61^{2} + \left(31 a + 58\right)\cdot 61^{3} + \left(26 a + 27\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 + 25\cdot 61 + 37\cdot 61^{2} + 40\cdot 61^{3} + 53\cdot 61^{4} +O(61^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |