Properties

Label 3.246016.4t4.a
Dimension $3$
Group $A_4$
Conductor $246016$
Indicator $1$

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:\(246016\)\(\medspace = 2^{8} \cdot 31^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.246016.2
Galois orbit size: $1$
Smallest permutation container: $A_4$
Parity: even
Projective image: $A_4$
Projective field: Galois closure of 4.0.246016.2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 3 + 13\cdot 23 + 13\cdot 23^{2} + 12\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 21\cdot 23 + 17\cdot 23^{2} + 4\cdot 23^{3} + 14\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 16\cdot 23 + 19\cdot 23^{2} + 23^{3} + 9\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 18 + 17\cdot 23 + 17\cdot 23^{2} + 3\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\) 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.