Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(22024249\)\(\medspace = 13^{2} \cdot 19^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.22024249.1 |
| 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 6.2.22024249.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{4} - 15x^{3} - 15x^{2} - 8x + 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 6 a + 33 + \left(70 a + 4\right)\cdot 73 + \left(62 a + 45\right)\cdot 73^{2} + \left(59 a + 29\right)\cdot 73^{3} + \left(25 a + 36\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 44\cdot 73 + 38\cdot 73^{2} + 28\cdot 73^{3} + 22\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 + 50\cdot 73 + 26\cdot 73^{2} + 22\cdot 73^{3} + 41\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 a + 14 + \left(58 a + 68\right)\cdot 73 + \left(71 a + 39\right)\cdot 73^{2} + \left(36 a + 49\right)\cdot 73^{3} + \left(22 a + 53\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 a + 49 + \left(14 a + 61\right)\cdot 73 + \left(a + 50\right)\cdot 73^{2} + \left(36 a + 15\right)\cdot 73^{3} + \left(50 a + 11\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 67 a + 51 + \left(2 a + 62\right)\cdot 73 + \left(10 a + 17\right)\cdot 73^{2} + 13 a\cdot 73^{3} + \left(47 a + 54\right)\cdot 73^{4} +O(73^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.