Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(501\!\cdots\!641\)\(\medspace = 41^{4} \cdot 64921^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.6.2661761.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T183 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.6.2661761.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 6x^{4} + 3x^{3} + 9x^{2} - x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 33 + 34\cdot 67 + 14\cdot 67^{2} + 25\cdot 67^{3} + 12\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 1 + \left(32 a + 56\right)\cdot 67 + \left(62 a + 10\right)\cdot 67^{2} + \left(50 a + 52\right)\cdot 67^{3} + \left(32 a + 5\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 + 43\cdot 67 + 33\cdot 67^{2} + 42\cdot 67^{3} + 36\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 a + 38 + \left(34 a + 25\right)\cdot 67 + \left(4 a + 27\right)\cdot 67^{2} + \left(16 a + 59\right)\cdot 67^{3} + \left(34 a + 18\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 20\cdot 67 + 8\cdot 67^{3} + 51\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 21\cdot 67 + 47\cdot 67^{2} + 13\cdot 67^{3} + 9\cdot 67^{4} +O(67^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $5$ | ✓ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ | |
| $15$ | $2$ | $(1,2)$ | $-3$ | |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ | |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ | |
| $40$ | $3$ | $(1,2,3)$ | $2$ | |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ | |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ | |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ | |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ | |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |