Properties

Label 3.349e2.4t4.1c1
Dimension 3
Group $A_4$
Conductor $ 349^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$121801= 349^{2} $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 10 x^{2} + 3 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 $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 9 + 5\cdot 37 + 27\cdot 37^{2} + 10\cdot 37^{3} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 28\cdot 37 + 22\cdot 37^{2} + 31\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 + 30\cdot 37 + 8\cdot 37^{2} + 7\cdot 37^{3} + 12\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 + 9\cdot 37 + 15\cdot 37^{2} + 24\cdot 37^{3} + 6\cdot 37^{4} +O\left(37^{ 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.