Properties

Label 2.499.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $499$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(499\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 3.1.499.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.499.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.499.1

Defining polynomial

$f(x)$$=$ \( x^{3} + 4x - 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 9 + 19\cdot 127 + 37\cdot 127^{2} + 38\cdot 127^{3} + 35\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 32 + 86\cdot 127 + 54\cdot 127^{2} + 97\cdot 127^{3} + 124\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 86 + 21\cdot 127 + 35\cdot 127^{2} + 118\cdot 127^{3} + 93\cdot 127^{4} +O(127^{5})\) Copy content 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.