Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(58589\)\(\medspace = 41 \cdot 1429 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.58589.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.58589.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.58589.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} + x^{3} - 4x^{2} + 4x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{2} + 96x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 67 a + 5 + \left(41 a + 60\right)\cdot 97 + \left(80 a + 34\right)\cdot 97^{2} + \left(12 a + 48\right)\cdot 97^{3} + 51 a\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 44 + \left(39 a + 3\right)\cdot 97 + \left(6 a + 81\right)\cdot 97^{2} + \left(2 a + 55\right)\cdot 97^{3} + \left(18 a + 32\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 74 + 72\cdot 97 + 36\cdot 97^{2} + 12\cdot 97^{3} + 55\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 72 + \left(55 a + 34\right)\cdot 97 + \left(16 a + 73\right)\cdot 97^{2} + \left(84 a + 77\right)\cdot 97^{3} + \left(45 a + 38\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 85 a + 56 + \left(57 a + 30\right)\cdot 97 + \left(90 a + 48\right)\cdot 97^{2} + \left(94 a + 51\right)\cdot 97^{3} + \left(78 a + 48\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 + 89\cdot 97 + 16\cdot 97^{2} + 45\cdot 97^{3} + 18\cdot 97^{4} +O(97^{5})\)
|
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 value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.