Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(3700\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 3.1.3700.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.3700.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 20\cdot 31 + 9\cdot 31^{2} + 13\cdot 31^{3} + 2\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 27 + 29\cdot 31 + 11\cdot 31^{2} + 23\cdot 31^{3} + 15\cdot 31^{4} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 30 + 11\cdot 31 + 9\cdot 31^{2} + 25\cdot 31^{3} + 12\cdot 31^{4} +O(31^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)$ | $0$ |
$2$ | $3$ | $(1,2,3)$ | $-1$ |