Properties

Label 2.588.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $588$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(588\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 3.1.588.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.588.1

Defining polynomial

$f(x)$$=$\(x^{3} - x^{2} + 5 x + 1\)  Toggle raw display.

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

Roots:
$r_{ 1 }$ $=$ \( 33 + 19\cdot 61 + 35\cdot 61^{2} + 7\cdot 61^{3} + 3\cdot 61^{4} +O(61^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 38 + 57\cdot 61 + 61^{2} + 34\cdot 61^{3} + 4\cdot 61^{4} +O(61^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 52 + 44\cdot 61 + 23\cdot 61^{2} + 19\cdot 61^{3} + 53\cdot 61^{4} +O(61^{5})\)  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.