Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(46757\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.46757.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.46757.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.46757.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{3} - 2x^{2} + 2x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 13 a + 7 + \left(61 a + 18\right)\cdot 97 + \left(73 a + 48\right)\cdot 97^{2} + \left(45 a + 91\right)\cdot 97^{3} + \left(35 a + 15\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 80 + 52\cdot 97 + 10\cdot 97^{2} + 73\cdot 97^{3} + 93\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 79 a + 82 + \left(29 a + 45\right)\cdot 97 + \left(50 a + 48\right)\cdot 97^{2} + \left(2 a + 11\right)\cdot 97^{3} + \left(42 a + 86\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 84 a + 20 + \left(35 a + 66\right)\cdot 97 + \left(23 a + 60\right)\cdot 97^{2} + \left(51 a + 63\right)\cdot 97^{3} + \left(61 a + 5\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 64 + \left(67 a + 93\right)\cdot 97 + \left(46 a + 68\right)\cdot 97^{2} + \left(94 a + 60\right)\cdot 97^{3} + \left(54 a + 28\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 + 14\cdot 97 + 54\cdot 97^{2} + 87\cdot 97^{3} + 60\cdot 97^{4} +O(97^{5})\)
|
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.