Properties

Label 2.327.3t2.a
Dimension $2$
Group $S_3$
Conductor $327$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:\(327\)\(\medspace = 3 \cdot 109 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 3.1.327.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.327.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 6 + 3\cdot 11 + 3\cdot 11^{2} + 9\cdot 11^{3} + 4\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 3\cdot 11 + 8\cdot 11^{2} + 8\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 9 + 3\cdot 11 + 10\cdot 11^{2} + 3\cdot 11^{3} +O(11^{5})\)  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.