Properties

Label 2.11_233.3t2.1
Dimension 2
Group $S_3$
Conductor $ 11 \cdot 233 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$2563= 11 \cdot 233 $
Artin number field: Splitting field of $f= x^{3} - x^{2} + x - 10 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 12 + 15\cdot 61 + 50\cdot 61^{2} + 36\cdot 61^{3} + 5\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 + 39\cdot 61 + 26\cdot 61^{2} + 14\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 27 + 6\cdot 61 + 45\cdot 61^{2} + 23\cdot 61^{3} + 41\cdot 61^{4} +O\left(61^{ 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.