Properties

Label 3.33489.4t4.a
Dimension $3$
Group $A_4$
Conductor $33489$
Indicator $1$

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:\(33489\)\(\medspace = 3^{2} \cdot 61^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.4.33489.1
Galois orbit size: $1$
Smallest permutation container: $A_4$
Parity: even
Projective image: $A_4$
Projective field: Galois closure of 4.4.33489.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 123 + 43\cdot 233 + 143\cdot 233^{2} + 102\cdot 233^{3} + 125\cdot 233^{4} +O(233^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 132 + 64\cdot 233 + 31\cdot 233^{2} + 176\cdot 233^{3} + 204\cdot 233^{4} +O(233^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 216 + 117\cdot 233 + 44\cdot 233^{2} + 179\cdot 233^{3} + 128\cdot 233^{4} +O(233^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 228 + 6\cdot 233 + 14\cdot 233^{2} + 8\cdot 233^{3} + 7\cdot 233^{4} +O(233^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character values
$c1$
$1$ $1$ $()$ $3$
$3$ $2$ $(1,2)(3,4)$ $-1$
$4$ $3$ $(1,2,3)$ $0$
$4$ $3$ $(1,3,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.