Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(19951\)\(\medspace = 71 \cdot 281 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.19951.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.19951.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.19951.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} + x^{3} - 3x^{2} + 3x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 198 + 203\cdot 467 + 185\cdot 467^{2} + 314\cdot 467^{3} + 409\cdot 467^{4} +O(467^{5})\)
$r_{ 2 }$ |
$=$ |
\( 338 + 186\cdot 467 + 79\cdot 467^{2} + 398\cdot 467^{3} + 270\cdot 467^{4} +O(467^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 433 + 369\cdot 467 + 305\cdot 467^{2} + 344\cdot 467^{3} + 48\cdot 467^{4} +O(467^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 437 + 388\cdot 467 + 81\cdot 467^{2} + 126\cdot 467^{3} + 349\cdot 467^{4} +O(467^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 464 + 251\cdot 467 + 281\cdot 467^{2} + 217\cdot 467^{3} + 322\cdot 467^{4} +O(467^{5})\)
| |
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 value |
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,2)$ | $2$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.