Properties

Label 3.2e6_3e4_5e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 3^{4} \cdot 5^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$129600= 2^{6} \cdot 3^{4} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{3} - 4 x^{2} - 6 x - 4 $ 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 $\Q_{ 467 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 49 + 50\cdot 467 + 28\cdot 467^{2} + 158\cdot 467^{3} + 139\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 59 + 367\cdot 467 + 148\cdot 467^{2} + 411\cdot 467^{3} + 177\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 102 + 228\cdot 467 + 80\cdot 467^{2} + 286\cdot 467^{3} + 364\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 319 + 331\cdot 467 + 385\cdot 467^{2} + 388\cdot 467^{3} + 209\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 405 + 423\cdot 467 + 290\cdot 467^{2} + 156\cdot 467^{3} + 42\cdot 467^{4} +O\left(467^{ 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}$
$12$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.