Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(162409\)\(\medspace = 13^{2} \cdot 31^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.156075049.1 |
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 5.1.156075049.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 31x^{2} + 31x + 31 \) . |
The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 57 + 167\cdot 467 + 58\cdot 467^{2} + 389\cdot 467^{3} + 407\cdot 467^{4} +O(467^{5})\) |
$r_{ 2 }$ | $=$ | \( 272 + 204\cdot 467 + 155\cdot 467^{2} + 14\cdot 467^{3} + 376\cdot 467^{4} +O(467^{5})\) |
$r_{ 3 }$ | $=$ | \( 308 + 20\cdot 467 + 320\cdot 467^{2} + 154\cdot 467^{3} + 164\cdot 467^{4} +O(467^{5})\) |
$r_{ 4 }$ | $=$ | \( 318 + 174\cdot 467 + 217\cdot 467^{2} + 306\cdot 467^{3} + 317\cdot 467^{4} +O(467^{5})\) |
$r_{ 5 }$ | $=$ | \( 446 + 366\cdot 467 + 182\cdot 467^{2} + 69\cdot 467^{3} + 135\cdot 467^{4} +O(467^{5})\) |
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.