Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(160000\)\(\medspace = 2^{8} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.100000000.1 |
| 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 5.1.100000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 10x^{3} - 20x^{2} + 10x + 216 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 5 + \left(9 a + 16\right)\cdot 19 + \left(18 a + 5\right)\cdot 19^{2} + \left(8 a + 8\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} + \left(11 a + 3\right)\cdot 19^{5} + \left(10 a + 5\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 19 + 4\cdot 19^{2} + 14\cdot 19^{3} + 5\cdot 19^{4} + 11\cdot 19^{5} + 18\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 9 + \left(12 a + 10\right)\cdot 19 + \left(6 a + 9\right)\cdot 19^{2} + \left(18 a + 2\right)\cdot 19^{3} + \left(7 a + 11\right)\cdot 19^{4} + \left(17 a + 2\right)\cdot 19^{5} + \left(4 a + 11\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 7 + \left(6 a + 6\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + 14\cdot 19^{3} + 11 a\cdot 19^{4} + \left(a + 12\right)\cdot 19^{5} + \left(14 a + 17\right)\cdot 19^{6} +O(19^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 8 + \left(9 a + 3\right)\cdot 19 + 15\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(13 a + 17\right)\cdot 19^{4} + \left(7 a + 8\right)\cdot 19^{5} + \left(8 a + 4\right)\cdot 19^{6} +O(19^{7})\)
|
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.