Properties

Label 5.50587.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 50587 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$50587 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - 2 x^{3} + 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.50587.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 271 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 271 }$: $ x^{2} + 269 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 110 + 20\cdot 271 + 12\cdot 271^{2} + 45\cdot 271^{3} + 18\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 242 a + 72 + \left(142 a + 93\right)\cdot 271 + \left(162 a + 47\right)\cdot 271^{2} + \left(47 a + 183\right)\cdot 271^{3} + \left(130 a + 142\right)\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 183 a + \left(199 a + 34\right)\cdot 271 + \left(33 a + 78\right)\cdot 271^{2} + \left(7 a + 83\right)\cdot 271^{3} + \left(5 a + 38\right)\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 88 a + 95 + \left(71 a + 250\right)\cdot 271 + \left(237 a + 216\right)\cdot 271^{2} + \left(263 a + 63\right)\cdot 271^{3} + \left(265 a + 41\right)\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 29 a + 14 + \left(128 a + 137\right)\cdot 271 + \left(108 a + 229\right)\cdot 271^{2} + \left(223 a + 115\right)\cdot 271^{3} + \left(140 a + 84\right)\cdot 271^{4} +O\left(271^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 252 + 6\cdot 271 + 229\cdot 271^{2} + 50\cdot 271^{3} + 217\cdot 271^{4} +O\left(271^{ 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.