Basic invariants
| Dimension: | $10$ |
| Group: | $A_6$ |
| Conductor: | \(23713122135310336\)\(\medspace = 2^{18} \cdot 67^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.287296.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Projective image: | $A_6$ |
| Projective field: | Galois closure of 6.2.287296.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 5 + \left(26 a + 29\right)\cdot 31 + \left(6 a + 1\right)\cdot 31^{2} + \left(20 a + 13\right)\cdot 31^{3} + \left(12 a + 4\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 28\cdot 31 + 22\cdot 31^{2} + 15\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 26 + \left(18 a + 4\right)\cdot 31 + \left(a + 22\right)\cdot 31^{2} + \left(6 a + 21\right)\cdot 31^{3} + \left(4 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 30 + \left(4 a + 23\right)\cdot 31 + \left(24 a + 19\right)\cdot 31^{2} + \left(10 a + 15\right)\cdot 31^{3} + \left(18 a + 9\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 13 + \left(12 a + 1\right)\cdot 31 + \left(29 a + 7\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(26 a + 6\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 + 5\cdot 31 + 19\cdot 31^{2} + 25\cdot 31^{3} + 30\cdot 31^{4} +O(31^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $10$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |