Basic invariants
Dimension: | $4$ |
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: | $S_5$ |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,2)$ | $-2$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |