Properties

Label 3.2e6_5e2_11e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 5^{2} \cdot 11^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$193600= 2^{6} \cdot 5^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{5} + 2 x^{3} - 2 x^{2} - x + 2 $ 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 $\Q_{ 337 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 58 + 230\cdot 337 + 27\cdot 337^{2} + 118\cdot 337^{3} + 147\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 94 + 170\cdot 337 + 335\cdot 337^{2} + 55\cdot 337^{3} + 34\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 98 + 291\cdot 337 + 288\cdot 337^{2} + 320\cdot 337^{3} + 120\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 166 + 167\cdot 337 + 336\cdot 337^{2} + 293\cdot 337^{3} + 303\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 258 + 151\cdot 337 + 22\cdot 337^{2} + 222\cdot 337^{3} + 67\cdot 337^{4} +O\left(337^{ 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.