Properties

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

Related objects

Learn more

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 number field: Galois closure of 3.1.588.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.588.1

Galois action

Roots of defining polynomial

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 values
$c1$
$1$ $1$ $()$ $2$
$3$ $2$ $(1,2)$ $0$
$2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.