Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(10509453369140625\)\(\medspace = 3^{16} \cdot 5^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.20503125.2 |
| 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 6.2.20503125.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 15x^{2} - 3x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 84 a + 32 + \left(61 a + 94\right)\cdot 97 + \left(29 a + 80\right)\cdot 97^{2} + \left(91 a + 22\right)\cdot 97^{3} + \left(81 a + 3\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 39 + \left(59 a + 92\right)\cdot 97 + \left(79 a + 25\right)\cdot 97^{2} + \left(78 a + 44\right)\cdot 97^{3} + \left(20 a + 18\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 75 a + 32 + \left(85 a + 93\right)\cdot 97 + \left(7 a + 34\right)\cdot 97^{2} + \left(16 a + 92\right)\cdot 97^{3} + \left(21 a + 57\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 10 + \left(11 a + 7\right)\cdot 97 + \left(89 a + 54\right)\cdot 97^{2} + \left(80 a + 3\right)\cdot 97^{3} + \left(75 a + 63\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 71 a + 65 + \left(37 a + 28\right)\cdot 97 + \left(17 a + 46\right)\cdot 97^{2} + \left(18 a + 43\right)\cdot 97^{3} + \left(76 a + 57\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 19 + \left(35 a + 72\right)\cdot 97 + \left(67 a + 48\right)\cdot 97^{2} + \left(5 a + 84\right)\cdot 97^{3} + \left(15 a + 90\right)\cdot 97^{4} +O(97^{5})\)
|
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.