Basic invariants
| Dimension: | $8$ |
| Group: | $A_6$ |
| Conductor: | \(15542770074624\)\(\medspace = 2^{10} \cdot 3^{12} \cdot 13^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.7884864.1 |
| Galois orbit size: | $2$ |
| 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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.