Properties

Label 3.2e4_31e2.4t4.1c1
Dimension 3
Group $A_4$
Conductor $ 2^{4} \cdot 31^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$15376= 2^{4} \cdot 31^{2} $
Artin number field: Splitting field of $f= x^{4} + 7 x^{2} - 2 x + 14 $ 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 $\Q_{ 151 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 60 + 26\cdot 151 + 95\cdot 151^{2} + 134\cdot 151^{3} + 112\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 62 + 80\cdot 151 + 36\cdot 151^{2} + 92\cdot 151^{3} + 81\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 80 + 24\cdot 151 + 148\cdot 151^{2} + 86\cdot 151^{3} + 107\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 100 + 19\cdot 151 + 22\cdot 151^{2} + 139\cdot 151^{3} + 150\cdot 151^{4} +O\left(151^{ 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.