Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(47666341533904\)\(\medspace = 2^{4} \cdot 14389^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.230224.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.14389.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.230224.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 64 + 276\cdot 311 + 188\cdot 311^{2} + 44\cdot 311^{3} + 34\cdot 311^{4} +O(311^{5})\) |
$r_{ 2 }$ | $=$ | \( 85 + 137\cdot 311 + 132\cdot 311^{2} + 298\cdot 311^{3} + 273\cdot 311^{4} +O(311^{5})\) |
$r_{ 3 }$ | $=$ | \( 119 + 155\cdot 311 + 303\cdot 311^{2} + 113\cdot 311^{3} + 291\cdot 311^{4} +O(311^{5})\) |
$r_{ 4 }$ | $=$ | \( 152 + 234\cdot 311 + 196\cdot 311^{2} + 266\cdot 311^{3} + 17\cdot 311^{4} +O(311^{5})\) |
$r_{ 5 }$ | $=$ | \( 204 + 129\cdot 311 + 111\cdot 311^{2} + 209\cdot 311^{3} + 4\cdot 311^{4} +O(311^{5})\) |
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.