Properties

Label 5.40543.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 40543 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$40543 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{4} - 3 x^{3} + 4 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.40543.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 21 a + 6 + \left(4 a + 21\right)\cdot 23 + \left(4 a + 17\right)\cdot 23^{2} + \left(16 a + 12\right)\cdot 23^{3} + \left(8 a + 4\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 2 + \left(18 a + 10\right)\cdot 23 + \left(18 a + 21\right)\cdot 23^{2} + \left(6 a + 17\right)\cdot 23^{3} + \left(14 a + 5\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 10 + \left(16 a + 19\right)\cdot 23 + \left(3 a + 17\right)\cdot 23^{2} + \left(9 a + 12\right)\cdot 23^{3} + \left(11 a + 19\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 + 11\cdot 23 + 22\cdot 23^{2} + 5\cdot 23^{3} + 4\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 11 + \left(6 a + 17\right)\cdot 23 + \left(19 a + 8\right)\cdot 23^{2} + \left(13 a + 4\right)\cdot 23^{3} + \left(11 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 + 12\cdot 23 + 3\cdot 23^{2} + 15\cdot 23^{3} + 23^{4} +O\left(23^{ 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.