Properties

Label 2.411.3t2.a
Dimension 2
Group $S_3$
Conductor $ 3 \cdot 137 $
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$411= 3 \cdot 137 $
Artin number field: Splitting field of $f= x^{3} - x^{2} + 5 x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.411.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 5 + 31\cdot 41 + 22\cdot 41^{2} + 25\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 + 29\cdot 41 + 6\cdot 41^{2} + 16\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 21\cdot 41 + 11\cdot 41^{2} + 40\cdot 41^{3} + 4\cdot 41^{4} +O\left(41^{ 5 }\right)$

Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $

Cycle notation
$(1,2,3)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ Character values
$c1$
$1$ $1$ $()$ $2$
$3$ $2$ $(1,2)$ $0$
$2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.