Properties

Label 3.2e6_3e4_7e2.4t4.2c1
Dimension 3
Group $A_4$
Conductor $ 2^{6} \cdot 3^{4} \cdot 7^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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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} - 12 x^{2} - 8 x + 72 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_4$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 9 + 10\cdot 23 + 18\cdot 23^{2} + 5\cdot 23^{3} + 21\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 8\cdot 23 + 12\cdot 23^{2} + 22\cdot 23^{3} + 17\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 9\cdot 23 + 14\cdot 23^{2} + 8\cdot 23^{3} + 3\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 + 16\cdot 23 + 9\cdot 23^{3} + 3\cdot 23^{4} +O\left(23^{ 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 value
$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.