Properties

Label 2.243.3t2.b.a
Dimension $2$
Group $S_3$
Conductor $243$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(243\)\(\medspace = 3^{5}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 3.1.243.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.243.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 47\cdot 61 + 43\cdot 61^{2} + 4\cdot 61^{3} + 39\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 26\cdot 61 + 49\cdot 61^{2} + 20\cdot 61^{3} + 32\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 52 + 48\cdot 61 + 28\cdot 61^{2} + 35\cdot 61^{3} + 50\cdot 61^{4} +O(61^{5})\) Copy content 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.