Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(1852754544393661\)\(\medspace = 263^{3} \cdot 467^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.122821.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | even |
Determinant: | 1.122821.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.122821.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 47 + 249\cdot 349 + 190\cdot 349^{2} + 89\cdot 349^{3} + 228\cdot 349^{4} +O(349^{5})\) |
$r_{ 2 }$ | $=$ | \( 106 + 124\cdot 349 + 205\cdot 349^{2} + 296\cdot 349^{3} + 191\cdot 349^{4} +O(349^{5})\) |
$r_{ 3 }$ | $=$ | \( 116 + 333\cdot 349 + 319\cdot 349^{2} + 166\cdot 349^{3} + 43\cdot 349^{4} +O(349^{5})\) |
$r_{ 4 }$ | $=$ | \( 181 + 168\cdot 349 + 25\cdot 349^{2} + 178\cdot 349^{3} + 226\cdot 349^{4} +O(349^{5})\) |
$r_{ 5 }$ | $=$ | \( 250 + 171\cdot 349 + 305\cdot 349^{2} + 315\cdot 349^{3} + 7\cdot 349^{4} +O(349^{5})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$10$ | $2$ | $(1,2)$ | $0$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.