Basic invariants
Dimension: | $5$ |
Group: | $A_6$ |
Conductor: | \(1971216\)\(\medspace = 2^{4} \cdot 3^{6} \cdot 13^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.7884864.1 |
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 6.2.7884864.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 6x^{3} - 6x^{2} - 6x - 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 52 + 18\cdot 73 + 13\cdot 73^{2} + 29\cdot 73^{3} + 32\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 42 a + 64 + \left(22 a + 13\right)\cdot 73 + \left(54 a + 41\right)\cdot 73^{2} + \left(5 a + 17\right)\cdot 73^{3} + \left(54 a + 68\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 31 a + 44 + \left(50 a + 39\right)\cdot 73 + \left(18 a + 35\right)\cdot 73^{2} + \left(67 a + 53\right)\cdot 73^{3} + \left(18 a + 5\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 12 a + 34 + \left(3 a + 49\right)\cdot 73 + \left(61 a + 62\right)\cdot 73^{2} + \left(28 a + 26\right)\cdot 73^{3} + \left(56 a + 25\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 28 + 50\cdot 73 + 42\cdot 73^{2} + 39\cdot 73^{3} + 67\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 61 a + 70 + \left(69 a + 46\right)\cdot 73 + \left(11 a + 23\right)\cdot 73^{2} + \left(44 a + 52\right)\cdot 73^{3} + \left(16 a + 19\right)\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $5$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$40$ | $3$ | $(1,2,3)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.