Properties

Label 4.3147076.5t4.a.a
Dimension $4$
Group $A_5$
Conductor $3147076$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $A_5$
Conductor: \(3147076\)\(\medspace = 2^{2} \cdot 887^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.3147076.1
Galois orbit size: $1$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: 5.1.3147076.1

Defining polynomial

$f(x)$$=$\(x^{5} - 2 x^{4} + 9 x^{3} - 12 x^{2} + 30 x + 2\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 38\cdot 181 + 181^{2} + 175\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 31\cdot 181 + 125\cdot 181^{2} + 20\cdot 181^{3} + 137\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 41 + 166\cdot 181 + 120\cdot 181^{2} + 9\cdot 181^{3} + 10\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 143 + 82\cdot 181 + 88\cdot 181^{2} + 177\cdot 181^{3} + 103\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 163 + 43\cdot 181 + 26\cdot 181^{2} + 160\cdot 181^{3} + 166\cdot 181^{4} +O(181^{5})\)  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$$1$$()$$4$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$12$$5$$(1,2,3,4,5)$$-1$
$12$$5$$(1,3,4,5,2)$$-1$

The blue line marks the conjugacy class containing complex conjugation.