Basic invariants
| Dimension: | $8$ |
| Group: | $A_6$ |
| Conductor: | \(38806720086016\)\(\medspace = 2^{18} \cdot 23^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.71639296.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Projective image: | $A_6$ |
| Projective field: | Galois closure of 6.2.71639296.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 641 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 210 + 349\cdot 641 + 367\cdot 641^{2} + 225\cdot 641^{3} + 591\cdot 641^{4} +O(641^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 292 + 506\cdot 641 + 51\cdot 641^{2} + 40\cdot 641^{3} + 75\cdot 641^{4} +O(641^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 367 + 327\cdot 641 + 583\cdot 641^{2} + 387\cdot 641^{3} + 364\cdot 641^{4} +O(641^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 493 + 420\cdot 641 + 417\cdot 641^{2} + 521\cdot 641^{3} + 72\cdot 641^{4} +O(641^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 599 + 165\cdot 641 + 550\cdot 641^{2} + 558\cdot 641^{3} + 188\cdot 641^{4} +O(641^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 604 + 152\cdot 641 + 593\cdot 641^{2} + 188\cdot 641^{3} + 630\cdot 641^{4} +O(641^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $8$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |