Properties

Label 5.17_2659.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 17 \cdot 2659 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$45203= 17 \cdot 2659 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} + x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.17_2659.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 16 a + 34 + \left(30 a + 27\right)\cdot 53 + \left(13 a + 48\right)\cdot 53^{2} + \left(23 a + 30\right)\cdot 53^{3} + \left(47 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 44 + 36\cdot 53 + 39\cdot 53^{2} + 11\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 37 a + 45 + \left(22 a + 26\right)\cdot 53 + \left(39 a + 19\right)\cdot 53^{2} + \left(29 a + 4\right)\cdot 53^{3} + \left(5 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 28 + 5\cdot 53 + 28\cdot 53^{2} + 26\cdot 53^{3} + 10\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 46 a + 45 + \left(44 a + 43\right)\cdot 53 + \left(31 a + 49\right)\cdot 53^{2} + \left(48 a + 40\right)\cdot 53^{3} + \left(40 a + 2\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 17 + \left(8 a + 18\right)\cdot 53 + \left(21 a + 26\right)\cdot 53^{2} + \left(4 a + 44\right)\cdot 53^{3} + \left(12 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.