Properties

Label 3.5e2_71e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 5^{2} \cdot 71^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$126025= 5^{2} \cdot 71^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 3 x^{3} - 4 x^{2} + 5 x - 1 $ 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 $\Q_{ 457 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 232 + 411\cdot 457 + 331\cdot 457^{2} + 218\cdot 457^{3} + 88\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 265 + 412\cdot 457 + 390\cdot 457^{2} + 376\cdot 457^{3} + 347\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 434 + 441\cdot 457 + 117\cdot 457^{2} + 448\cdot 457^{3} + 125\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 444 + 338\cdot 457 + 281\cdot 457^{2} + 19\cdot 457^{3} + 138\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 454 + 222\cdot 457 + 248\cdot 457^{2} + 307\cdot 457^{3} + 213\cdot 457^{4} +O\left(457^{ 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} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.