Properties

Label 3.2e6_19e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$23104= 2^{6} \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 8 x^{3} - 32 x^{2} + 45 x - 17 $ 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 $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 14\cdot 17 + 12\cdot 17^{2} + 6\cdot 17^{3} + 14\cdot 17^{4} + 16\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 16 + 7\cdot 17 + \left(15 a + 10\right)\cdot 17^{2} + \left(5 a + 12\right)\cdot 17^{3} + \left(10 a + 7\right)\cdot 17^{4} + \left(a + 5\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 12 + \left(5 a + 5\right)\cdot 17 + \left(16 a + 4\right)\cdot 17^{2} + \left(3 a + 3\right)\cdot 17^{3} + \left(4 a + 8\right)\cdot 17^{4} + \left(3 a + 16\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 8 a + 8 + \left(16 a + 16\right)\cdot 17 + \left(a + 7\right)\cdot 17^{2} + \left(11 a + 3\right)\cdot 17^{3} + \left(6 a + 12\right)\cdot 17^{4} + \left(15 a + 13\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 13 a + 16 + \left(11 a + 6\right)\cdot 17 + 15\cdot 17^{2} + \left(13 a + 7\right)\cdot 17^{3} + \left(12 a + 8\right)\cdot 17^{4} + \left(13 a + 15\right)\cdot 17^{5} +O\left(17^{ 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.