Properties

Label 2.76.3t2.a
Dimension 2
Group $S_3$
Conductor $ 2^{2} \cdot 19 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$76= 2^{2} \cdot 19 $
Artin number field: Splitting field of $f= x^{3} - 2 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 3.1.76.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 12 + 4\cdot 23 + 8\cdot 23^{2} + 22\cdot 23^{3} + 10\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 14 + 6\cdot 23 + 5\cdot 23^{2} + 7\cdot 23^{3} + 15\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 11\cdot 23 + 9\cdot 23^{2} + 16\cdot 23^{3} + 19\cdot 23^{4} +O\left(23^{ 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.