Properties

Label 3.5e2_89e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 5^{2} \cdot 89^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$198025= 5^{2} \cdot 89^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 5 x^{3} - x^{2} + 6 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 19 + 60\cdot 137 + 117\cdot 137^{2} + 111\cdot 137^{3} + 42\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 24 + 13\cdot 137 + 60\cdot 137^{2} + 118\cdot 137^{3} + 89\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 74 + 9\cdot 137 + 52\cdot 137^{2} + 86\cdot 137^{3} + 25\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 78 + 131\cdot 137 + 16\cdot 137^{2} + 132\cdot 137^{3} + 40\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 80 + 59\cdot 137 + 27\cdot 137^{2} + 99\cdot 137^{3} + 74\cdot 137^{4} +O\left(137^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2,3)$
$(3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.