Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(693\!\cdots\!801\)\(\medspace = 17^{4} \cdot 9547^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.162299.1 |
| 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 6.0.162299.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + 3x^{2} - x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 20 a + 56 + \left(90 a + 77\right)\cdot 97 + \left(13 a + 88\right)\cdot 97^{2} + \left(70 a + 36\right)\cdot 97^{3} + \left(23 a + 89\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 + 33\cdot 97 + 53\cdot 97^{2} + 66\cdot 97^{3} + 48\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 13 + \left(65 a + 17\right)\cdot 97 + \left(72 a + 25\right)\cdot 97^{2} + 79 a\cdot 97^{3} + \left(10 a + 56\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 + 68\cdot 97 + 78\cdot 97^{2} + 86\cdot 97^{3} + 66\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 a + 52 + \left(31 a + 43\right)\cdot 97 + \left(24 a + 32\right)\cdot 97^{2} + \left(17 a + 7\right)\cdot 97^{3} + \left(86 a + 84\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 a + 76 + \left(6 a + 50\right)\cdot 97 + \left(83 a + 12\right)\cdot 97^{2} + \left(26 a + 93\right)\cdot 97^{3} + \left(73 a + 42\right)\cdot 97^{4} +O(97^{5})\)
|
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.