Basic invariants
| Dimension: | $10$ |
| Group: | $A_6$ |
| Conductor: | \(8792367498140625\)\(\medspace = 3^{14} \cdot 5^{6} \cdot 7^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.22325625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_6$ |
| Projective stem field: | Galois closure of 6.2.22325625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 3x^{4} - 12x^{2} - 15x + 16 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 8\cdot 31 + 20\cdot 31^{2} + 27\cdot 31^{3} + 2\cdot 31^{4} + 26\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 7 + \left(6 a + 23\right)\cdot 31 + \left(15 a + 7\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + 23 a\cdot 31^{4} + \left(14 a + 25\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 7\cdot 31 + 29\cdot 31^{2} + 14\cdot 31^{3} + 20\cdot 31^{4} + 9\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 10 + \left(8 a + 7\right)\cdot 31 + \left(a + 5\right)\cdot 31^{2} + \left(17 a + 26\right)\cdot 31^{3} + \left(6 a + 17\right)\cdot 31^{4} + \left(18 a + 16\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 14 + \left(24 a + 17\right)\cdot 31 + 15 a\cdot 31^{2} + \left(20 a + 1\right)\cdot 31^{3} + \left(7 a + 6\right)\cdot 31^{4} + 16 a\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 2 + \left(22 a + 29\right)\cdot 31 + \left(29 a + 29\right)\cdot 31^{2} + \left(13 a + 27\right)\cdot 31^{3} + \left(24 a + 13\right)\cdot 31^{4} + \left(12 a + 15\right)\cdot 31^{5} +O(31^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.