Properties

Label 5.33791.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 33791 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$33791 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} + x^{3} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.33791.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 5 a + \left(24 a + 44\right)\cdot 47 + \left(6 a + 7\right)\cdot 47^{2} + \left(27 a + 32\right)\cdot 47^{3} + 24\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 42 a + 10 + \left(22 a + 40\right)\cdot 47 + \left(40 a + 43\right)\cdot 47^{2} + \left(19 a + 32\right)\cdot 47^{3} + \left(46 a + 45\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 a + 6 + \left(24 a + 37\right)\cdot 47 + \left(28 a + 10\right)\cdot 47^{2} + \left(14 a + 32\right)\cdot 47^{3} + \left(7 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 41 + 30\cdot 47 + 11\cdot 47^{2} + 16\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 9 + \left(22 a + 14\right)\cdot 47 + \left(18 a + 43\right)\cdot 47^{2} + \left(32 a + 32\right)\cdot 47^{3} + \left(39 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 29 + 21\cdot 47 + 23\cdot 47^{2} + 41\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 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.