Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(12712961507219\)\(\medspace = 23339^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 5.3.23339.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Projective image: | $S_5$ |
Projective field: | Galois closure of 5.3.23339.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 14 + 240\cdot 337 + 231\cdot 337^{2} + 234\cdot 337^{3} + 190\cdot 337^{4} +O(337^{5})\)
$r_{ 2 }$ |
$=$ |
\( 17 + 172\cdot 337 + 42\cdot 337^{2} + 18\cdot 337^{3} + 286\cdot 337^{4} +O(337^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 180 + 52\cdot 337 + 192\cdot 337^{2} + 200\cdot 337^{3} + 336\cdot 337^{4} +O(337^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 203 + 332\cdot 337 + 260\cdot 337^{2} + 4\cdot 337^{3} + 51\cdot 337^{4} +O(337^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 261 + 213\cdot 337 + 283\cdot 337^{2} + 215\cdot 337^{3} + 146\cdot 337^{4} +O(337^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,2)$ | $-2$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |