Basic invariants
Dimension: | $8$ |
Group: | $A_6$ |
Conductor: | \(2025170913606201\)\(\medspace = 3^{16} \cdot 19^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.95004009.3 |
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 6.2.95004009.3 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} - 3x^{4} + 8x^{3} + 9x^{2} - 9x + 6 \) . |
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 11 + 42\cdot 311 + 169\cdot 311^{2} + 275\cdot 311^{3} + 188\cdot 311^{4} +O(311^{5})\) |
$r_{ 2 }$ | $=$ | \( 178 + 66\cdot 311 + 83\cdot 311^{2} + 306\cdot 311^{3} + 283\cdot 311^{4} +O(311^{5})\) |
$r_{ 3 }$ | $=$ | \( 224 + 261\cdot 311 + 143\cdot 311^{2} + 74\cdot 311^{3} + 94\cdot 311^{4} +O(311^{5})\) |
$r_{ 4 }$ | $=$ | \( 261 + 310\cdot 311 + 19\cdot 311^{2} + 227\cdot 311^{3} + 110\cdot 311^{4} +O(311^{5})\) |
$r_{ 5 }$ | $=$ | \( 271 + 309\cdot 311 + 283\cdot 311^{2} + 257\cdot 311^{3} + 5\cdot 311^{4} +O(311^{5})\) |
$r_{ 6 }$ | $=$ | \( 302 + 252\cdot 311 + 232\cdot 311^{2} + 102\cdot 311^{3} + 249\cdot 311^{4} +O(311^{5})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$45$ | $2$ | $(1,2)(3,4)$ | $0$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.