Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(77841\)\(\medspace = 3^{4} \cdot 31^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.74805201.1 |
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 5.1.74805201.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + 19x^{3} - 41x^{2} + 32x - 5 \) . |
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 37 + 102\cdot 389 + 312\cdot 389^{2} + 343\cdot 389^{3} + 14\cdot 389^{4} +O(389^{5})\)
$r_{ 2 }$ |
$=$ |
\( 45 + 385\cdot 389 + 204\cdot 389^{2} + 157\cdot 389^{3} + 111\cdot 389^{4} +O(389^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 72 + 135\cdot 389 + 59\cdot 389^{2} + 47\cdot 389^{3} + 249\cdot 389^{4} +O(389^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 86 + 332\cdot 389 + 226\cdot 389^{2} + 362\cdot 389^{3} + 331\cdot 389^{4} +O(389^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 150 + 212\cdot 389 + 363\cdot 389^{2} + 255\cdot 389^{3} + 70\cdot 389^{4} +O(389^{5})\)
| |
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.