Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(263169\)\(\medspace = 3^{6} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.263169.1 |
| 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 5.1.263169.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + x^{3} + 2x^{2} + x + 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 13\cdot 43 + 34\cdot 43^{2} + 31\cdot 43^{3} + 12\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 a + 23 + \left(30 a + 9\right)\cdot 43 + \left(26 a + 38\right)\cdot 43^{2} + \left(14 a + 9\right)\cdot 43^{3} + \left(14 a + 4\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 20 + \left(3 a + 30\right)\cdot 43 + \left(15 a + 26\right)\cdot 43^{2} + \left(5 a + 6\right)\cdot 43^{3} + \left(17 a + 5\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 16 + \left(12 a + 4\right)\cdot 43 + \left(16 a + 34\right)\cdot 43^{2} + \left(28 a + 40\right)\cdot 43^{3} + \left(28 a + 3\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 a + 25 + \left(39 a + 28\right)\cdot 43 + \left(27 a + 38\right)\cdot 43^{2} + \left(37 a + 39\right)\cdot 43^{3} + \left(25 a + 16\right)\cdot 43^{4} +O(43^{5})\)
|
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.