Properties

Label 5.20627.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 20627 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$20627 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - 2 x^{3} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.20627.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $ x^{2} + 274 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 62 a + 226 + \left(43 a + 116\right)\cdot 277 + \left(152 a + 9\right)\cdot 277^{2} + \left(231 a + 84\right)\cdot 277^{3} + \left(175 a + 85\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 230 + 135\cdot 277 + 119\cdot 277^{2} + 146\cdot 277^{3} + 55\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 118 + 184\cdot 277 + 194\cdot 277^{2} + 65\cdot 277^{3} + 153\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 41 + 146\cdot 277 + 30\cdot 277^{2} + 9\cdot 277^{3} + 24\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 215 a + 135 + \left(233 a + 184\right)\cdot 277 + \left(124 a + 145\right)\cdot 277^{2} + \left(45 a + 72\right)\cdot 277^{3} + \left(101 a + 104\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 82 + 63\cdot 277 + 54\cdot 277^{2} + 176\cdot 277^{3} + 131\cdot 277^{4} +O\left(277^{ 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.