Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(129\!\cdots\!000\)\(\medspace = 2^{12} \cdot 3^{12} \cdot 5^{24} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.25312500.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1252 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.25312500.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 5x^{3} + 12x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 10 + \left(3 a + 8\right)\cdot 17 + \left(10 a + 5\right)\cdot 17^{2} + 6\cdot 17^{3} + \left(8 a + 12\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 16 + \left(9 a + 1\right)\cdot 17 + \left(13 a + 10\right)\cdot 17^{2} + \left(6 a + 12\right)\cdot 17^{3} + \left(3 a + 16\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 1 + \left(7 a + 13\right)\cdot 17 + \left(a + 7\right)\cdot 17^{2} + \left(8 a + 11\right)\cdot 17^{3} + \left(a + 14\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 4 + \left(13 a + 1\right)\cdot 17 + \left(6 a + 12\right)\cdot 17^{2} + \left(16 a + 13\right)\cdot 17^{3} + \left(8 a + 2\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 14 + \left(7 a + 13\right)\cdot 17 + \left(3 a + 13\right)\cdot 17^{2} + \left(10 a + 5\right)\cdot 17^{3} + \left(13 a + 13\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 9 + \left(9 a + 12\right)\cdot 17 + \left(15 a + 1\right)\cdot 17^{2} + \left(8 a + 1\right)\cdot 17^{3} + \left(15 a + 8\right)\cdot 17^{4} +O(17^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.