Properties

Label 6.3e3_3253e3.20t35.1c1
Dimension 6
Group $S_5$
Conductor $ 3^{3} \cdot 3253^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_5$
Conductor:$929428432479= 3^{3} \cdot 3253^{3} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 20T35
Parity: Odd
Determinant: 1.3_3253.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 12 + \left(a + 10\right)\cdot 19 + \left(12 a + 18\right)\cdot 19^{2} + \left(5 a + 16\right)\cdot 19^{3} + \left(16 a + 1\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 11\cdot 19 + 15\cdot 19^{2} + 15\cdot 19^{3} + 15\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 a + \left(16 a + 12\right)\cdot 19 + \left(7 a + 10\right)\cdot 19^{2} + \left(15 a + 12\right)\cdot 19^{3} + \left(a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 15 + \left(2 a + 13\right)\cdot 19 + \left(11 a + 1\right)\cdot 19^{2} + \left(3 a + 1\right)\cdot 19^{3} + \left(17 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 15 + \left(17 a + 8\right)\cdot 19 + \left(6 a + 10\right)\cdot 19^{2} + \left(13 a + 10\right)\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$6$
$10$$2$$(1,2)$$0$
$15$$2$$(1,2)(3,4)$$-2$
$20$$3$$(1,2,3)$$0$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$1$
$20$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.