Properties

Label 2.3_137.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 3 \cdot 137 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$411= 3 \cdot 137 $
Artin number field: Splitting field of $f= x^{3} - x^{2} + 5 x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Odd
Determinant: 1.3_137.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 5 + 31\cdot 41 + 22\cdot 41^{2} + 25\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 2 } &= 13 + 29\cdot 41 + 6\cdot 41^{2} + 16\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 3 } &= 24 + 21\cdot 41 + 11\cdot 41^{2} + 40\cdot 41^{3} + 4\cdot 41^{4} +O\left(41^{ 5 }\right) \\ \end{aligned}\]

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.