Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(40000\)\(\medspace = 2^{6} \cdot 5^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.25000000.3 |
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 5.1.25000000.3 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} + 10x^{3} - 40x^{2} + 60x - 32 \) . |
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:
$r_{ 1 }$ | $=$ | \( 6 a + 10 + \left(24 a + 21\right)\cdot 53 + \left(24 a + 20\right)\cdot 53^{2} + \left(8 a + 51\right)\cdot 53^{3} + \left(41 a + 30\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 2 }$ | $=$ | \( 23 a + 39 + \left(29 a + 19\right)\cdot 53 + \left(12 a + 24\right)\cdot 53^{2} + \left(21 a + 8\right)\cdot 53^{3} + \left(39 a + 36\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 3 }$ | $=$ | \( 30 a + 25 + \left(23 a + 8\right)\cdot 53 + \left(40 a + 45\right)\cdot 53^{2} + \left(31 a + 27\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 4 }$ | $=$ | \( 51 + 50\cdot 53 + 27\cdot 53^{2} + 10\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\) |
$r_{ 5 }$ | $=$ | \( 47 a + 34 + \left(28 a + 5\right)\cdot 53 + \left(28 a + 41\right)\cdot 53^{2} + \left(44 a + 7\right)\cdot 53^{3} + \left(11 a + 28\right)\cdot 53^{4} +O(53^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.