Properties

Label 5.89_509.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 89 \cdot 509 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$45301= 89 \cdot 509 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.89_509.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 2 a + 8 + \left(3 a + 205\right)\cdot 277 + \left(145 a + 227\right)\cdot 277^{2} + \left(170 a + 240\right)\cdot 277^{3} + \left(139 a + 186\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 232 a + 121 + \left(208 a + 203\right)\cdot 277 + \left(147 a + 104\right)\cdot 277^{2} + \left(240 a + 32\right)\cdot 277^{3} + \left(73 a + 182\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 275 a + 14 + \left(273 a + 212\right)\cdot 277 + \left(131 a + 105\right)\cdot 277^{2} + \left(106 a + 53\right)\cdot 277^{3} + \left(137 a + 158\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 45 a + 263 + \left(68 a + 43\right)\cdot 277 + \left(129 a + 62\right)\cdot 277^{2} + \left(36 a + 52\right)\cdot 277^{3} + \left(203 a + 163\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 90 a + 78 + \left(86 a + 137\right)\cdot 277 + \left(107 a + 47\right)\cdot 277^{2} + \left(161 a + 176\right)\cdot 277^{3} + \left(244 a + 199\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 187 a + 71 + \left(190 a + 29\right)\cdot 277 + \left(169 a + 6\right)\cdot 277^{2} + \left(115 a + 276\right)\cdot 277^{3} + \left(32 a + 217\right)\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.