Properties

Label 2.643.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $643$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(643\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 3.1.643.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.643.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.643.1

Defining polynomial

$f(x)$$=$\(x^{3} - 2 x - 5\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 11 + 51\cdot 163 + 22\cdot 163^{2} + 146\cdot 163^{3} + 103\cdot 163^{4} +O(163^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 55 + 63\cdot 163 + 143\cdot 163^{2} + 14\cdot 163^{3} + 27\cdot 163^{4} +O(163^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 97 + 48\cdot 163 + 160\cdot 163^{2} + 163^{3} + 32\cdot 163^{4} +O(163^{5})\)  Toggle raw display

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 value
$1$$1$$()$$2$
$3$$2$$(1,2)$$0$
$2$$3$$(1,2,3)$$-1$

The blue line marks the conjugacy class containing complex conjugation.