Properties

Label 3.2e6_41e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 41^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$107584= 2^{6} \cdot 41^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 16 x^{3} + 36 x^{2} + 267 x - 319 $ 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 $\Q_{ 67 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 3 + 20\cdot 67 + 55\cdot 67^{2} + 61\cdot 67^{3} + 46\cdot 67^{4} + 56\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 28 + 53\cdot 67 + 47\cdot 67^{2} + 14\cdot 67^{3} + 9\cdot 67^{4} + 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 49 + 62\cdot 67 + 53\cdot 67^{2} + 7\cdot 67^{3} + 11\cdot 67^{4} + 55\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 56 + 21\cdot 67 + 25\cdot 67^{2} + 32\cdot 67^{3} + 11\cdot 67^{4} + 22\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 66 + 42\cdot 67 + 18\cdot 67^{2} + 17\cdot 67^{3} + 55\cdot 67^{4} + 65\cdot 67^{5} +O\left(67^{ 6 }\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.