Properties

Label 2.331.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 331 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$331 $
Artin number field: Splitting field of $f= x^{3} - x^{2} + 3 x - 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Odd
Determinant: 1.331.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 23 + 42\cdot 83 + 67\cdot 83^{2} + 58\cdot 83^{3} + 80\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 + 70\cdot 83 + 81\cdot 83^{2} + 66\cdot 83^{3} + 54\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 33 + 53\cdot 83 + 16\cdot 83^{2} + 40\cdot 83^{3} + 30\cdot 83^{4} +O\left(83^{ 5 }\right)$

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.