Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(188356\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.181010116.1 |
| 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 5.1.181010116.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 12x^{3} - 10x^{2} + 63x + 119 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15\cdot 17 + 10\cdot 17^{2} + 6\cdot 17^{3} + 14\cdot 17^{4} + 14\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 10 + \left(5 a + 4\right)\cdot 17 + \left(9 a + 7\right)\cdot 17^{2} + \left(12 a + 3\right)\cdot 17^{3} + \left(10 a + 2\right)\cdot 17^{4} + 10 a\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 2 + 16\cdot 17 + \left(8 a + 6\right)\cdot 17^{2} + \left(5 a + 1\right)\cdot 17^{3} + \left(13 a + 13\right)\cdot 17^{4} + \left(15 a + 16\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 9 + \left(16 a + 9\right)\cdot 17 + \left(8 a + 14\right)\cdot 17^{2} + \left(11 a + 15\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} + \left(a + 2\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 14 + \left(11 a + 5\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(4 a + 6\right)\cdot 17^{3} + 6 a\cdot 17^{4} + 6 a\cdot 17^{5} +O(17^{6})\)
|
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.