Properties

Label 2.562636.3t2.a
Dimension $2$
Group $S_3$
Conductor $562636$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:\(562636\)\(\medspace = 2^{2} \cdot 140659 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 3.1.562636.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.562636.1

Galois action

Roots of defining polynomial

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