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 5.1.1000000.2 |
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 5.1.1000000.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} + 20x - 16 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a + 18 + \left(4 a + 11\right)\cdot 23 + \left(7 a + 19\right)\cdot 23^{2} + \left(14 a + 18\right)\cdot 23^{3} + 21 a\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 a + 17 + \left(21 a + 6\right)\cdot 23 + \left(19 a + 17\right)\cdot 23^{2} + \left(16 a + 16\right)\cdot 23^{3} + \left(22 a + 17\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 21 a + 22 + \left(18 a + 17\right)\cdot 23 + \left(15 a + 6\right)\cdot 23^{2} + \left(8 a + 17\right)\cdot 23^{3} + \left(a + 6\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 4 }$ | $=$ | \( 17 + 17\cdot 23 + 12\cdot 23^{2} + 8\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\) |
$r_{ 5 }$ | $=$ | \( 11 a + 18 + \left(a + 14\right)\cdot 23 + \left(3 a + 12\right)\cdot 23^{2} + \left(6 a + 7\right)\cdot 23^{3} +O(23^{5})\) |
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.