Properties

Label 2.1620.3t2.b.a
Dimension $2$
Group $S_3$
Conductor $1620$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 3.1.1620.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.20.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.1620.1

Defining polynomial

$f(x)$$=$\(x^{3} - 3 x - 8\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 11 + 25\cdot 43 + 12\cdot 43^{2} + 23\cdot 43^{3} + 10\cdot 43^{4} +O(43^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 13 + 40\cdot 43 + 29\cdot 43^{2} + 16\cdot 43^{3} + 10\cdot 43^{4} +O(43^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 20\cdot 43 + 3\cdot 43^{3} + 22\cdot 43^{4} +O(43^{5})\)  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.