Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(2888816545234944\)\(\medspace = 2^{26} \cdot 3^{16} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.497664.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.497664.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + x^{2} - 2x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 88 a + 69 + \left(96 a + 34\right)\cdot 97 + \left(71 a + 70\right)\cdot 97^{2} + \left(73 a + 14\right)\cdot 97^{3} + \left(64 a + 51\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 60 + 43\cdot 97 + \left(25 a + 45\right)\cdot 97^{2} + \left(23 a + 16\right)\cdot 97^{3} + \left(32 a + 42\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 62 a + 10 + \left(77 a + 3\right)\cdot 97 + \left(31 a + 33\right)\cdot 97^{2} + \left(93 a + 42\right)\cdot 97^{3} + \left(41 a + 81\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 + 96\cdot 97 + 37\cdot 97^{2} + 86\cdot 97^{3} + 19\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a + 72 + \left(19 a + 18\right)\cdot 97 + \left(65 a + 84\right)\cdot 97^{2} + \left(3 a + 6\right)\cdot 97^{3} + \left(55 a + 30\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 + 94\cdot 97 + 19\cdot 97^{2} + 27\cdot 97^{3} + 66\cdot 97^{4} +O(97^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.