Properties

Label 3.2e6_3e4_7e2.4t4.3
Dimension 3
Group $A_4$
Conductor $ 2^{6} \cdot 3^{4} \cdot 7^{2}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$254016= 2^{6} \cdot 3^{4} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{4} + 18 x^{2} - 8 x + 120 $ 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 $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4 + 29\cdot 79 + 11\cdot 79^{2} + 60\cdot 79^{3} + 61\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 45 + 43\cdot 79 + 4\cdot 79^{2} + 53\cdot 79^{3} + 65\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 50 + 3\cdot 79 + 34\cdot 79^{2} + 67\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 59 + 2\cdot 79 + 29\cdot 79^{2} + 56\cdot 79^{3} + 71\cdot 79^{4} +O\left(79^{ 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.