Properties

Label 5.43063.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 43063 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$43063 $
Artin number field: Splitting field of $f= x^{6} - x^{3} + x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.43063.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 3 + 61\cdot 89 + 17\cdot 89^{2} + 6\cdot 89^{3} + 40\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 71 a + 24 + \left(26 a + 33\right)\cdot 89 + \left(82 a + 59\right)\cdot 89^{2} + \left(49 a + 16\right)\cdot 89^{3} + \left(11 a + 75\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 44 + 72\cdot 89 + 53\cdot 89^{2} + 30\cdot 89^{3} + 2\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 71 + 33\cdot 89 + 27\cdot 89^{2} + 7\cdot 89^{3} + 79\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 49 + 5\cdot 89 + 34\cdot 89^{2} + 11\cdot 89^{3} + 53\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 18 a + 76 + \left(62 a + 60\right)\cdot 89 + \left(6 a + 74\right)\cdot 89^{2} + \left(39 a + 16\right)\cdot 89^{3} + \left(77 a + 17\right)\cdot 89^{4} +O\left(89^{ 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.