Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(1960\)\(\medspace = 2^{3} \cdot 5 \cdot 7^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 3.1.1960.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.1960.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 9 + 2\cdot 59 + 58\cdot 59^{2} + 22\cdot 59^{3} + 42\cdot 59^{4} +O(59^{5})\) |
$r_{ 2 }$ | $=$ | \( 19 + 29\cdot 59 + 5\cdot 59^{2} + 4\cdot 59^{3} + 24\cdot 59^{4} +O(59^{5})\) |
$r_{ 3 }$ | $=$ | \( 32 + 27\cdot 59 + 54\cdot 59^{2} + 31\cdot 59^{3} + 51\cdot 59^{4} +O(59^{5})\) |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)$ | $0$ |
$2$ | $3$ | $(1,2,3)$ | $-1$ |