Properties

Label 3.129600.12t33.a.a
Dimension $3$
Group $A_5$
Conductor $129600$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(129600\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.129600.1
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 5.1.129600.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{3} - 4x^{2} - 6x - 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 49 + 50\cdot 467 + 28\cdot 467^{2} + 158\cdot 467^{3} + 139\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 59 + 367\cdot 467 + 148\cdot 467^{2} + 411\cdot 467^{3} + 177\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 102 + 228\cdot 467 + 80\cdot 467^{2} + 286\cdot 467^{3} + 364\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 319 + 331\cdot 467 + 385\cdot 467^{2} + 388\cdot 467^{3} + 209\cdot 467^{4} +O(467^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 405 + 423\cdot 467 + 290\cdot 467^{2} + 156\cdot 467^{3} + 42\cdot 467^{4} +O(467^{5})\) 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.