Properties

Label 2.648.3t2.b
Dimension $2$
Group $S_3$
Conductor $648$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:\(648\)\(\medspace = 2^{3} \cdot 3^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 3.1.648.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.648.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 15 + 78\cdot 83 + 19\cdot 83^{2} + 19\cdot 83^{3} + 78\cdot 83^{4} +O(83^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 75 + 57\cdot 83 + 51\cdot 83^{2} + 26\cdot 83^{3} + 75\cdot 83^{4} +O(83^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 76 + 29\cdot 83 + 11\cdot 83^{2} + 37\cdot 83^{3} + 12\cdot 83^{4} +O(83^{5})\)  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.