Properties

Label 3.2e4_3e4_7e2.4t4.1c1
Dimension 3
Group $A_4$
Conductor $ 2^{4} \cdot 3^{4} \cdot 7^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$63504= 2^{4} \cdot 3^{4} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} - 4 x + 20 $ 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 $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 24 + 8\cdot 59 + 23\cdot 59^{2} + 12\cdot 59^{3} + 32\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 31 + 5\cdot 59 + 27\cdot 59^{2} + 51\cdot 59^{3} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 32 + 59 + 28\cdot 59^{2} + 26\cdot 59^{3} + 53\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 33 + 43\cdot 59 + 39\cdot 59^{2} + 27\cdot 59^{3} + 31\cdot 59^{4} +O\left(59^{ 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.