Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(14254694598461563\)\(\medspace = 242467^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.242467.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | odd |
| Determinant: | 1.242467.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.4.242467.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} - x^{3} - 2x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 30\cdot 53 + 11\cdot 53^{2} + 29\cdot 53^{3} + 48\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 a + 9 + \left(15 a + 23\right)\cdot 53 + \left(14 a + 44\right)\cdot 53^{2} + 20 a\cdot 53^{3} + \left(28 a + 24\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 + 29\cdot 53 + 25\cdot 53^{2} + 18\cdot 53^{3} + 52\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 46 + 5\cdot 53 + 15\cdot 53^{2} + 36\cdot 53^{3} + 41\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 14 + \left(37 a + 45\right)\cdot 53 + \left(38 a + 32\right)\cdot 53^{2} + \left(32 a + 14\right)\cdot 53^{3} + \left(24 a + 11\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 24\cdot 53 + 29\cdot 53^{2} + 6\cdot 53^{3} + 34\cdot 53^{4} +O(53^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $9$ | |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ | |
| $15$ | $2$ | $(1,2)$ | $3$ | ✓ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ | |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | |
| $40$ | $3$ | $(1,2,3)$ | $0$ | |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ | |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ | |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ | |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |