Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(36100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 19^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.13032100.1 |
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 5.1.13032100.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 11x^{3} + 6x^{2} + 64x - 74 \) . |
The roots of $f$ are computed in $\Q_{ 557 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 14 + 251\cdot 557 + 288\cdot 557^{2} + 375\cdot 557^{3} + 218\cdot 557^{4} +O(557^{5})\) |
$r_{ 2 }$ | $=$ | \( 235 + 435\cdot 557 + 247\cdot 557^{2} + 321\cdot 557^{3} + 239\cdot 557^{4} +O(557^{5})\) |
$r_{ 3 }$ | $=$ | \( 375 + 224\cdot 557 + 116\cdot 557^{2} + 75\cdot 557^{3} + 92\cdot 557^{4} +O(557^{5})\) |
$r_{ 4 }$ | $=$ | \( 506 + 262\cdot 557 + 255\cdot 557^{2} + 489\cdot 557^{3} + 304\cdot 557^{4} +O(557^{5})\) |
$r_{ 5 }$ | $=$ | \( 542 + 496\cdot 557 + 205\cdot 557^{2} + 409\cdot 557^{3} + 258\cdot 557^{4} +O(557^{5})\) |
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.