Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(202500\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.126562500.1 |
| 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 5.1.126562500.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 15x^{3} - 65x^{2} - 90x - 57 \)
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The roots of $f$ are computed in $\Q_{ 487 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 128 + 250\cdot 487 + 389\cdot 487^{2} + 431\cdot 487^{3} + 390\cdot 487^{4} +O(487^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 186 + 407\cdot 487 + 388\cdot 487^{2} + 261\cdot 487^{3} + 146\cdot 487^{4} +O(487^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 291 + 264\cdot 487 + 269\cdot 487^{2} + 279\cdot 487^{3} + 412\cdot 487^{4} +O(487^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 421 + 213\cdot 487 + 229\cdot 487^{2} + 153\cdot 487^{3} + 76\cdot 487^{4} +O(487^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 435 + 324\cdot 487 + 183\cdot 487^{2} + 334\cdot 487^{3} + 434\cdot 487^{4} +O(487^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.