Basic invariants
Dimension: | $4$ |
Group: | $A_5$ |
Conductor: | \(8340544\)\(\medspace = 2^{6} \cdot 19^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.8340544.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_5$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_5$ |
Projective stem field: | Galois closure of 5.1.8340544.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + 8x^{3} - 32x^{2} + 45x - 17 \) . |
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:
$r_{ 1 }$ | $=$ | \( 14\cdot 17 + 12\cdot 17^{2} + 6\cdot 17^{3} + 14\cdot 17^{4} + 16\cdot 17^{5} +O(17^{6})\) |
$r_{ 2 }$ | $=$ | \( 9 a + 16 + 7\cdot 17 + \left(15 a + 10\right)\cdot 17^{2} + \left(5 a + 12\right)\cdot 17^{3} + \left(10 a + 7\right)\cdot 17^{4} + \left(a + 5\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 3 }$ | $=$ | \( 4 a + 12 + \left(5 a + 5\right)\cdot 17 + \left(16 a + 4\right)\cdot 17^{2} + \left(3 a + 3\right)\cdot 17^{3} + \left(4 a + 8\right)\cdot 17^{4} + \left(3 a + 16\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 4 }$ | $=$ | \( 8 a + 8 + \left(16 a + 16\right)\cdot 17 + \left(a + 7\right)\cdot 17^{2} + \left(11 a + 3\right)\cdot 17^{3} + \left(6 a + 12\right)\cdot 17^{4} + \left(15 a + 13\right)\cdot 17^{5} +O(17^{6})\) |
$r_{ 5 }$ | $=$ | \( 13 a + 16 + \left(11 a + 6\right)\cdot 17 + 15\cdot 17^{2} + \left(13 a + 7\right)\cdot 17^{3} + \left(12 a + 8\right)\cdot 17^{4} + \left(13 a + 15\right)\cdot 17^{5} +O(17^{6})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.