Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(7695324729\)\(\medspace = 3^{10} \cdot 19^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.95004009.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Projective image: | $A_6$ |
| Projective field: | Galois closure of 6.2.95004009.3 |
Galois action
Roots of defining polynomial
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |