Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(3656158440062976\)\(\medspace = 2^{20} \cdot 3^{20} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.7558272.2 |
| 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 6.2.7558272.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 3x^{4} - 4x^{3} - 6x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 10 + \left(38 a + 30\right)\cdot 67 + \left(38 a + 23\right)\cdot 67^{2} + \left(9 a + 51\right)\cdot 67^{3} + \left(19 a + 23\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 49 a + 60 + \left(2 a + 65\right)\cdot 67 + \left(17 a + 22\right)\cdot 67^{2} + \left(53 a + 27\right)\cdot 67^{3} + \left(55 a + 58\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 66 + \left(15 a + 25\right)\cdot 67 + \left(61 a + 49\right)\cdot 67^{2} + \left(52 a + 49\right)\cdot 67^{3} + \left(31 a + 63\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 a + 26 + \left(28 a + 44\right)\cdot 67 + \left(28 a + 5\right)\cdot 67^{2} + \left(57 a + 51\right)\cdot 67^{3} + \left(47 a + 23\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 55 + \left(64 a + 27\right)\cdot 67 + \left(49 a + 21\right)\cdot 67^{2} + \left(13 a + 22\right)\cdot 67^{3} + \left(11 a + 27\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 54 a + 51 + \left(51 a + 6\right)\cdot 67 + \left(5 a + 11\right)\cdot 67^{2} + \left(14 a + 66\right)\cdot 67^{3} + \left(35 a + 3\right)\cdot 67^{4} +O(67^{5})\)
|
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.