Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(18574970416\)\(\medspace = 2^{4} \cdot 1051^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.16816.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.1051.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.16816.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{3} - 2x^{2} - 4x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 541 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 72 + 518\cdot 541 + 520\cdot 541^{2} + 441\cdot 541^{3} + 281\cdot 541^{4} +O(541^{5})\) |
$r_{ 2 }$ | $=$ | \( 87 + 63\cdot 541 + 166\cdot 541^{2} + 541^{3} + 35\cdot 541^{4} +O(541^{5})\) |
$r_{ 3 }$ | $=$ | \( 246 + 91\cdot 541 + 97\cdot 541^{2} + 177\cdot 541^{3} + 148\cdot 541^{4} +O(541^{5})\) |
$r_{ 4 }$ | $=$ | \( 284 + 379\cdot 541 + 21\cdot 541^{2} + 301\cdot 541^{3} + 189\cdot 541^{4} +O(541^{5})\) |
$r_{ 5 }$ | $=$ | \( 393 + 29\cdot 541 + 276\cdot 541^{2} + 160\cdot 541^{3} + 427\cdot 541^{4} +O(541^{5})\) |
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.