Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.5856400.1 |
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 5.1.5856400.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} + 6x^{3} - 20x^{2} + 4x - 32 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ | \( 7 a + 13 + \left(30 a + 15\right)\cdot 41 + \left(7 a + 30\right)\cdot 41^{2} + \left(a + 27\right)\cdot 41^{3} + \left(14 a + 11\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 29 a + 33 + \left(27 a + 31\right)\cdot 41 + \left(36 a + 39\right)\cdot 41^{2} + \left(9 a + 37\right)\cdot 41^{3} + \left(3 a + 24\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 12 a + 38 + \left(13 a + 3\right)\cdot 41 + \left(4 a + 40\right)\cdot 41^{2} + \left(31 a + 30\right)\cdot 41^{3} + \left(37 a + 24\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 34 a + 34 + \left(10 a + 16\right)\cdot 41 + \left(33 a + 23\right)\cdot 41^{2} + \left(39 a + 23\right)\cdot 41^{3} + \left(26 a + 11\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 7 + 14\cdot 41 + 30\cdot 41^{2} + 2\cdot 41^{3} + 9\cdot 41^{4} +O(41^{5})\) |
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.