Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(198025\)\(\medspace = 5^{2} \cdot 89^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.198025.1 |
| 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 5.1.198025.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 5x^{3} - x^{2} + 6x + 1 \)
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The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 60\cdot 137 + 117\cdot 137^{2} + 111\cdot 137^{3} + 42\cdot 137^{4} +O(137^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 24 + 13\cdot 137 + 60\cdot 137^{2} + 118\cdot 137^{3} + 89\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 74 + 9\cdot 137 + 52\cdot 137^{2} + 86\cdot 137^{3} + 25\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 131\cdot 137 + 16\cdot 137^{2} + 132\cdot 137^{3} + 40\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 80 + 59\cdot 137 + 27\cdot 137^{2} + 99\cdot 137^{3} + 74\cdot 137^{4} +O(137^{5})\)
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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.