Properties

Label 2.3e5.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 3^{5}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$243= 3^{5} $
Artin number field: Splitting field of $f= x^{3} - 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Odd
Determinant: 1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 4 + 47\cdot 61 + 43\cdot 61^{2} + 4\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right) \\ r_{ 2 } &= 5 + 26\cdot 61 + 49\cdot 61^{2} + 20\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right) \\ r_{ 3 } &= 52 + 48\cdot 61 + 28\cdot 61^{2} + 35\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right) \\ \end{aligned}\]

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.