Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(18496\)\(\medspace = 2^{6} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.18496.1 |
| 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 5.1.18496.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 2x^{2} - 2x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a + 33 + \left(14 a + 1\right)\cdot 37 + \left(34 a + 24\right)\cdot 37^{2} + \left(7 a + 27\right)\cdot 37^{3} + \left(6 a + 21\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 37 + 4\cdot 37^{2} + 24\cdot 37^{3} + 2\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 20 + \left(29 a + 30\right)\cdot 37 + \left(30 a + 31\right)\cdot 37^{2} + \left(25 a + 17\right)\cdot 37^{3} + \left(13 a + 9\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 a + 16 + \left(22 a + 16\right)\cdot 37 + \left(2 a + 36\right)\cdot 37^{2} + \left(29 a + 24\right)\cdot 37^{3} + \left(30 a + 1\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 35 + \left(7 a + 23\right)\cdot 37 + \left(6 a + 14\right)\cdot 37^{2} + \left(11 a + 16\right)\cdot 37^{3} + \left(23 a + 1\right)\cdot 37^{4} +O(37^{5})\)
|
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.