Properties

Label 2.2e3_13.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 2^{3} \cdot 13 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$104= 2^{3} \cdot 13 $
Artin number field: Splitting field of $f= x^{3} - x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Odd
Determinant: 1.2e3_13.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 14 + 13\cdot 31 + 8\cdot 31^{3} + 19\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 2 } &= 21 + 13\cdot 31 + 30\cdot 31^{2} + 14\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 3 } &= 27 + 3\cdot 31 + 8\cdot 31^{3} + 20\cdot 31^{4} +O\left(31^{ 5 }\right) \\ \end{aligned}\]

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.