Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(245025\)\(\medspace = 3^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.245025.2 |
| 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 5.1.245025.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 3x^{3} - 2x^{2} + 5x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 46 + \left(5 a + 11\right)\cdot 53 + \left(20 a + 5\right)\cdot 53^{2} + 31 a\cdot 53^{3} + \left(30 a + 47\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 a + 41 + \left(33 a + 5\right)\cdot 53 + \left(45 a + 47\right)\cdot 53^{2} + \left(38 a + 16\right)\cdot 53^{3} + \left(46 a + 44\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 a + 9 + \left(47 a + 28\right)\cdot 53 + \left(32 a + 27\right)\cdot 53^{2} + \left(21 a + 52\right)\cdot 53^{3} + \left(22 a + 31\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a + 23 + \left(19 a + 11\right)\cdot 53 + \left(7 a + 37\right)\cdot 53^{2} + \left(14 a + 20\right)\cdot 53^{3} + \left(6 a + 33\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 41 + 48\cdot 53 + 41\cdot 53^{2} + 15\cdot 53^{3} + 2\cdot 53^{4} +O(53^{5})\)
|
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.