Basic invariants
| Dimension: | $6$ |
| Group: | $S_5$ |
| Conductor: | \(5515586099981504\)\(\medspace = 2^{6} \cdot 44171^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.5.176684.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | even |
| Projective image: | $S_5$ |
| Projective field: | Galois closure of 5.5.176684.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 + 133\cdot 269 + 44\cdot 269^{2} + 33\cdot 269^{3} + 60\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 136 + 243\cdot 269 + 187\cdot 269^{2} + 260\cdot 269^{3} + 125\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 176 + 64\cdot 269 + 77\cdot 269^{2} + 41\cdot 269^{3} + 77\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 224 + 37\cdot 269 + 111\cdot 269^{2} + 8\cdot 269^{3} + 49\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 248 + 58\cdot 269 + 117\cdot 269^{2} + 194\cdot 269^{3} + 225\cdot 269^{4} +O(269^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |