Properties

Label 3.2e2_13e2_17e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{2} \cdot 13^{2} \cdot 17^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$195364= 2^{2} \cdot 13^{2} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - 3 x^{3} + 9 x^{2} - 4 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 $\Q_{ 359 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 74 + 217\cdot 359 + 92\cdot 359^{2} + 249\cdot 359^{3} + 325\cdot 359^{4} +O\left(359^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 94 + 258\cdot 359 + 325\cdot 359^{2} + 96\cdot 359^{3} + 148\cdot 359^{4} +O\left(359^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 141 + 320\cdot 359 + 357\cdot 359^{2} + 25\cdot 359^{3} + 85\cdot 359^{4} +O\left(359^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 189 + 300\cdot 359 + 263\cdot 359^{2} + 278\cdot 359^{3} + 48\cdot 359^{4} +O\left(359^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 222 + 339\cdot 359 + 36\cdot 359^{2} + 67\cdot 359^{3} + 110\cdot 359^{4} +O\left(359^{ 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.