Properties

Label 3.61504.12t33.a.a
Dimension $3$
Group $A_5$
Conductor $61504$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(61504\)\(\medspace = 2^{6} \cdot 31^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.59105344.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: Galois closure of 5.1.59105344.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} + 14x^{3} - 18x^{2} + 47x - 36 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 54 + 21\cdot 257 + 142\cdot 257^{2} + 138\cdot 257^{3} + 153\cdot 257^{4} +O(257^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 90 + 102\cdot 257 + 212\cdot 257^{2} + 217\cdot 257^{3} + 210\cdot 257^{4} +O(257^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 107 + 6\cdot 257 + 204\cdot 257^{2} + 210\cdot 257^{3} + 181\cdot 257^{4} +O(257^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 117 + 113\cdot 257 + 152\cdot 257^{2} + 253\cdot 257^{3} + 223\cdot 257^{4} +O(257^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 148 + 13\cdot 257 + 60\cdot 257^{2} + 207\cdot 257^{3} +O(257^{5})\) Copy content 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}$
$12$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.