Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(144400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 19^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.144400.1 |
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 5.1.144400.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + 3x^{3} - 3x^{2} + 5x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 12\cdot 37 + 36\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\) |
$r_{ 2 }$ | $=$ | \( 11 a + 24 + \left(11 a + 30\right)\cdot 37 + \left(24 a + 12\right)\cdot 37^{2} + \left(20 a + 13\right)\cdot 37^{3} + \left(25 a + 11\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 3 }$ | $=$ | \( 29 a + 25 + \left(3 a + 8\right)\cdot 37 + \left(13 a + 12\right)\cdot 37^{2} + \left(8 a + 3\right)\cdot 37^{3} + \left(3 a + 2\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 4 }$ | $=$ | \( 8 a + 30 + \left(33 a + 31\right)\cdot 37 + \left(23 a + 23\right)\cdot 37^{2} + \left(28 a + 23\right)\cdot 37^{3} + \left(33 a + 6\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 5 }$ | $=$ | \( 26 a + 31 + \left(25 a + 27\right)\cdot 37 + \left(12 a + 24\right)\cdot 37^{2} + \left(16 a + 34\right)\cdot 37^{3} + \left(11 a + 18\right)\cdot 37^{4} +O(37^{5})\) |
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.