Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(134225699004481536\)\(\medspace = 2^{18} \cdot 3^{4} \cdot 43^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.355008.1 |
| 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 6.2.355008.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 2x^{3} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 36\cdot 61 + 30\cdot 61^{2} + 53\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 17\cdot 61 + 14\cdot 61^{2} + 23\cdot 61^{3} + 43\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 30 + \left(a + 45\right)\cdot 61 + \left(56 a + 12\right)\cdot 61^{2} + \left(58 a + 16\right)\cdot 61^{3} + \left(30 a + 49\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a + 46 + \left(59 a + 30\right)\cdot 61 + \left(4 a + 6\right)\cdot 61^{2} + \left(2 a + 19\right)\cdot 61^{3} + \left(30 a + 21\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 33 + \left(55 a + 5\right)\cdot 61 + \left(53 a + 60\right)\cdot 61^{2} + \left(6 a + 58\right)\cdot 61^{3} + \left(60 a + 51\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 48 a + 46 + \left(5 a + 47\right)\cdot 61 + \left(7 a + 58\right)\cdot 61^{2} + \left(54 a + 11\right)\cdot 61^{3} + 44\cdot 61^{4} +O(61^{5})\)
|
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.