Properties

Label 5.53_607.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 53 \cdot 607 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$32171= 53 \cdot 607 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - x^{3} + x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.53_607.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 20 a + 33 + \left(8 a + 9\right)\cdot 47 + \left(21 a + 38\right)\cdot 47^{2} + 9\cdot 47^{3} + \left(24 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 a + 41 + \left(42 a + 29\right)\cdot 47 + \left(5 a + 43\right)\cdot 47^{2} + \left(7 a + 42\right)\cdot 47^{3} + \left(a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 27 a + 26 + \left(38 a + 6\right)\cdot 47 + \left(25 a + 25\right)\cdot 47^{2} + \left(46 a + 36\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 16 + 3\cdot 47 + 39\cdot 47^{2} + 17\cdot 47^{3} + 31\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 30 a + 28 + \left(4 a + 3\right)\cdot 47 + \left(41 a + 13\right)\cdot 47^{2} + \left(39 a + 4\right)\cdot 47^{3} + \left(45 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 45 + 40\cdot 47 + 28\cdot 47^{2} + 29\cdot 47^{3} + 8\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.