Properties

Label 3.4338889.12t33.a.b
Dimension $3$
Group $A_5$
Conductor $4338889$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(4338889\)\(\medspace = 2083^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.4338889.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: 5.1.4338889.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} + 5 x^{3} + 11 x^{2} + 4 x - 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 241 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 47 + 194\cdot 241 + 5\cdot 241^{2} + 200\cdot 241^{3} + 188\cdot 241^{4} +O(241^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 88 + 84\cdot 241 + 45\cdot 241^{2} + 31\cdot 241^{3} + 45\cdot 241^{4} +O(241^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 175 + 27\cdot 241 + 12\cdot 241^{2} + 65\cdot 241^{3} + 218\cdot 241^{4} +O(241^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 188 + 88\cdot 241 + 233\cdot 241^{2} + 186\cdot 241^{3} + 5\cdot 241^{4} +O(241^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 226 + 86\cdot 241 + 185\cdot 241^{2} + 239\cdot 241^{3} + 23\cdot 241^{4} +O(241^{5})\)  Toggle raw display

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.