Basic invariants
Dimension: | $8$ |
Group: | $A_6$ |
Conductor: | \(38806720086016\)\(\medspace = 2^{18} \cdot 23^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.71639296.1 |
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 6.2.71639296.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{4} + 4x^{3} - 6x^{2} + 2x + 2 \) . |
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 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.