Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(111\!\cdots\!216\)\(\medspace = 2^{28} \cdot 3779^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.241856.1 |
| 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 6.2.241856.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - 4x^{3} + 5x^{2} - 4x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 19 + \left(19 a + 7\right)\cdot 71 + \left(39 a + 50\right)\cdot 71^{2} + \left(8 a + 40\right)\cdot 71^{3} + \left(47 a + 37\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 66 a + 9 + \left(42 a + 66\right)\cdot 71 + \left(48 a + 11\right)\cdot 71^{2} + \left(45 a + 44\right)\cdot 71^{3} + \left(25 a + 44\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 a + 14 + \left(51 a + 13\right)\cdot 71 + \left(31 a + 38\right)\cdot 71^{2} + \left(62 a + 18\right)\cdot 71^{3} + \left(23 a + 52\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 70 + \left(28 a + 14\right)\cdot 71 + \left(22 a + 66\right)\cdot 71^{2} + \left(25 a + 15\right)\cdot 71^{3} + \left(45 a + 50\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 15\cdot 71 + 70\cdot 71^{2} + 21\cdot 71^{3} + 49\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 + 25\cdot 71 + 47\cdot 71^{2} + 50\cdot 71^{4} +O(71^{5})\)
|
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.