Properties

Label 3.11e2_19e2.4t4.1c1
Dimension 3
Group $A_4$
Conductor $ 11^{2} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$43681= 11^{2} \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 9 x^{2} + 6 x + 25 $ 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 $\Q_{ 107 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 33 + 35\cdot 107 + 82\cdot 107^{2} + 36\cdot 107^{3} + 72\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 40 + 83\cdot 107 + 107^{2} + 86\cdot 107^{3} + 44\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 41 + 36\cdot 107 + 52\cdot 107^{2} + 94\cdot 107^{3} + 99\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 101 + 58\cdot 107 + 77\cdot 107^{2} + 103\cdot 107^{3} + 103\cdot 107^{4} +O\left(107^{ 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.