Properties

Label 3.2e6_5e4.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 5^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$40000= 2^{6} \cdot 5^{4} $
Artin number field: Splitting field of $f= x^{5} + 10 x^{3} - 40 x^{2} + 60 x - 32 $ 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 an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 6 a + 10 + \left(24 a + 21\right)\cdot 53 + \left(24 a + 20\right)\cdot 53^{2} + \left(8 a + 51\right)\cdot 53^{3} + \left(41 a + 30\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 a + 39 + \left(29 a + 19\right)\cdot 53 + \left(12 a + 24\right)\cdot 53^{2} + \left(21 a + 8\right)\cdot 53^{3} + \left(39 a + 36\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 30 a + 25 + \left(23 a + 8\right)\cdot 53 + \left(40 a + 45\right)\cdot 53^{2} + \left(31 a + 27\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 51 + 50\cdot 53 + 27\cdot 53^{2} + 10\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 47 a + 34 + \left(28 a + 5\right)\cdot 53 + \left(28 a + 41\right)\cdot 53^{2} + \left(44 a + 7\right)\cdot 53^{3} + \left(11 a + 28\right)\cdot 53^{4} +O\left(53^{ 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.