Properties

Label 3.3e2_61e2.4t4.1
Dimension 3
Group $A_4$
Conductor $ 3^{2} \cdot 61^{2}$
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$33489= 3^{2} \cdot 61^{2} $
Artin number field: Splitting field of $f= x^{4} - 7 x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_4$
Parity: Even

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\left(233^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 132 + 64\cdot 233 + 31\cdot 233^{2} + 176\cdot 233^{3} + 204\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 216 + 117\cdot 233 + 44\cdot 233^{2} + 179\cdot 233^{3} + 128\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 228 + 6\cdot 233 + 14\cdot 233^{2} + 8\cdot 233^{3} + 7\cdot 233^{4} +O\left(233^{ 5 }\right)$

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.