Properties

Label 3.2e8_5e4.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{8} \cdot 5^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$160000= 2^{8} \cdot 5^{4} $
Artin number field: Splitting field of $f= x^{5} - 10 x^{3} - 20 x^{2} + 10 x + 216 $ 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 $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 5 + \left(9 a + 16\right)\cdot 19 + \left(18 a + 5\right)\cdot 19^{2} + \left(8 a + 8\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} + \left(11 a + 3\right)\cdot 19^{5} + \left(10 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 9 + 19 + 4\cdot 19^{2} + 14\cdot 19^{3} + 5\cdot 19^{4} + 11\cdot 19^{5} + 18\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 17 a + 9 + \left(12 a + 10\right)\cdot 19 + \left(6 a + 9\right)\cdot 19^{2} + \left(18 a + 2\right)\cdot 19^{3} + \left(7 a + 11\right)\cdot 19^{4} + \left(17 a + 2\right)\cdot 19^{5} + \left(4 a + 11\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 7 + \left(6 a + 6\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + 14\cdot 19^{3} + 11 a\cdot 19^{4} + \left(a + 12\right)\cdot 19^{5} + \left(14 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 8 + \left(9 a + 3\right)\cdot 19 + 15\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(13 a + 17\right)\cdot 19^{4} + \left(7 a + 8\right)\cdot 19^{5} + \left(8 a + 4\right)\cdot 19^{6} +O\left(19^{ 7 }\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.