Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(149\!\cdots\!625\)\(\medspace = 5^{6} \cdot 9923^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.49615.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T145 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.49615.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{4} - x^{3} + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 54 + 17\cdot 71 + 11\cdot 71^{2} + 26\cdot 71^{3} + 23\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 56\cdot 71 + 61\cdot 71^{2} + 21\cdot 71^{3} + 44\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 65\cdot 71 + 39\cdot 71^{2} + 37\cdot 71^{3} + 35\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 1 + \left(41 a + 45\right)\cdot 71 + \left(25 a + 3\right)\cdot 71^{2} + \left(3 a + 39\right)\cdot 71^{3} + \left(26 a + 47\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 + 44\cdot 71 + 11\cdot 71^{2} + 68\cdot 71^{3} + 36\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 70 a + 3 + \left(29 a + 55\right)\cdot 71 + \left(45 a + 13\right)\cdot 71^{2} + \left(67 a + 20\right)\cdot 71^{3} + \left(44 a + 25\right)\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.