Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 3.1.756.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.756.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 11\cdot 17 + 7\cdot 17^{2} + 15\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 + 10\cdot 17^{2} + 14\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 + 5\cdot 17 + 16\cdot 17^{2} + 3\cdot 17^{3} + 10\cdot 17^{4} +O(17^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)$ | $0$ |
$2$ | $3$ | $(1,2,3)$ | $-1$ |