Properties

Label 3.5e2_13e2.4t4.1c1
Dimension 3
Group $A_4$
Conductor $ 5^{2} \cdot 13^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$4225= 5^{2} \cdot 13^{2} $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 3 x + 4 $ 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 $\Q_{ 157 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 7 + 5\cdot 157 + 128\cdot 157^{2} + 124\cdot 157^{3} + 60\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 120\cdot 157 + 100\cdot 157^{2} + 63\cdot 157^{3} + 28\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 46 + 59\cdot 157 + 102\cdot 157^{2} + 77\cdot 157^{3} + 69\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 96 + 129\cdot 157 + 139\cdot 157^{2} + 47\cdot 157^{3} + 155\cdot 157^{4} +O\left(157^{ 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.