Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(153424\)\(\medspace = 2^{4} \cdot 43 \cdot 223 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.153424.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.9589.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.153424.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} - 4x^{3} + 8x^{2} - 2 \)
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The roots of $f$ are computed in $\Q_{ 571 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 38 + 184\cdot 571 + 300\cdot 571^{2} + 297\cdot 571^{3} + 112\cdot 571^{4} +O(571^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 358 + 364\cdot 571 + 173\cdot 571^{2} + 340\cdot 571^{3} + 203\cdot 571^{4} +O(571^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 399 + 394\cdot 571 + 557\cdot 571^{2} + 363\cdot 571^{3} + 514\cdot 571^{4} +O(571^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 412 + 212\cdot 571 + 558\cdot 571^{2} + 475\cdot 571^{3} + 185\cdot 571^{4} +O(571^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 508 + 556\cdot 571 + 122\cdot 571^{2} + 235\cdot 571^{3} + 125\cdot 571^{4} +O(571^{5})\)
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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.