Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(133225\)\(\medspace = 5^{2} \cdot 73^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.133225.1 |
| 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 5.1.133225.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + x^{2} + 4x - 5 \)
|
The roots of $f$ are computed in $\Q_{ 401 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 270\cdot 401 + 222\cdot 401^{2} + 85\cdot 401^{3} + 317\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 92 + 174\cdot 401 + 2\cdot 401^{2} + 357\cdot 401^{3} + 342\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 181 + 304\cdot 401 + 218\cdot 401^{2} + 26\cdot 401^{3} + 231\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 191 + 11\cdot 401 + 187\cdot 401^{2} + 250\cdot 401^{3} + 93\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 329 + 41\cdot 401 + 171\cdot 401^{2} + 82\cdot 401^{3} + 218\cdot 401^{4} +O(401^{5})\)
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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.