Basic invariants
Dimension: | $5$ |
Group: | $A_6$ |
Conductor: | \(23066015625\)\(\medspace = 3^{10} \cdot 5^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.2562890625.1 |
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 6.2.2562890625.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} + 15x^{4} - 25x^{3} - 45x + 60 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{2} + 82x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 75 a + 87 + \left(5 a + 44\right)\cdot 89 + \left(56 a + 10\right)\cdot 89^{2} + \left(53 a + 31\right)\cdot 89^{3} + \left(75 a + 1\right)\cdot 89^{4} + \left(52 a + 13\right)\cdot 89^{5} +O(89^{6})\) |
$r_{ 2 }$ | $=$ | \( 3 a + 45 + \left(22 a + 45\right)\cdot 89 + \left(20 a + 25\right)\cdot 89^{2} + \left(72 a + 17\right)\cdot 89^{3} + \left(69 a + 77\right)\cdot 89^{4} + \left(31 a + 83\right)\cdot 89^{5} +O(89^{6})\) |
$r_{ 3 }$ | $=$ | \( 86 a + 66 + \left(66 a + 18\right)\cdot 89 + \left(68 a + 56\right)\cdot 89^{2} + \left(16 a + 57\right)\cdot 89^{3} + \left(19 a + 48\right)\cdot 89^{4} + \left(57 a + 58\right)\cdot 89^{5} +O(89^{6})\) |
$r_{ 4 }$ | $=$ | \( 68 + 87\cdot 89 + 21\cdot 89^{2} + 12\cdot 89^{3} + 4\cdot 89^{4} + 38\cdot 89^{5} +O(89^{6})\) |
$r_{ 5 }$ | $=$ | \( 14 a + 78 + \left(83 a + 10\right)\cdot 89 + \left(32 a + 41\right)\cdot 89^{2} + \left(35 a + 83\right)\cdot 89^{3} + \left(13 a + 31\right)\cdot 89^{4} + \left(36 a + 40\right)\cdot 89^{5} +O(89^{6})\) |
$r_{ 6 }$ | $=$ | \( 15 + 59\cdot 89 + 22\cdot 89^{2} + 65\cdot 89^{3} + 14\cdot 89^{4} + 33\cdot 89^{5} +O(89^{6})\) |
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 |
$1$ | $1$ | $()$ | $5$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$40$ | $3$ | $(1,2,3)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.