Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(69696\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.8433216.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.8433216.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 4x^{3} - 8x^{2} + 25x - 1 \)
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The roots of $f$ are computed in $\Q_{ 449 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 107 + 396\cdot 449 + 197\cdot 449^{2} + 434\cdot 449^{3} + 397\cdot 449^{4} +O(449^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 133 + 215\cdot 449 + 174\cdot 449^{2} + 346\cdot 449^{3} + 332\cdot 449^{4} +O(449^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 140 + 24\cdot 449 + 378\cdot 449^{2} + 324\cdot 449^{3} + 233\cdot 449^{4} +O(449^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 190 + 17\cdot 449 + 398\cdot 449^{2} + 425\cdot 449^{3} + 262\cdot 449^{4} +O(449^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 329 + 244\cdot 449 + 198\cdot 449^{2} + 264\cdot 449^{3} + 119\cdot 449^{4} +O(449^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.