Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(287296\)\(\medspace = 2^{6} \cdot 67^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.287296.1 |
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 5.1.287296.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} + 7x^{2} - 9x + 5 \) . |
The roots of $f$ are computed in $\Q_{ 263 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 53 + 203\cdot 263 + 22\cdot 263^{2} + 129\cdot 263^{3} + 229\cdot 263^{4} +O(263^{5})\) |
$r_{ 2 }$ | $=$ | \( 57 + 137\cdot 263 + 38\cdot 263^{2} + 116\cdot 263^{3} + 57\cdot 263^{4} +O(263^{5})\) |
$r_{ 3 }$ | $=$ | \( 80 + 215\cdot 263 + 78\cdot 263^{2} + 82\cdot 263^{3} + 224\cdot 263^{4} +O(263^{5})\) |
$r_{ 4 }$ | $=$ | \( 99 + 211\cdot 263 + 143\cdot 263^{2} + 253\cdot 263^{3} + 128\cdot 263^{4} +O(263^{5})\) |
$r_{ 5 }$ | $=$ | \( 238 + 21\cdot 263 + 242\cdot 263^{2} + 207\cdot 263^{3} + 148\cdot 263^{4} +O(263^{5})\) |
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.