Properties

Label 5.11_4241.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 11 \cdot 4241 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$46651= 11 \cdot 4241 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - x^{3} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.11_4241.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 8 + 15\cdot 61 + 30\cdot 61^{2} + 46\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 29 + \left(19 a + 58\right)\cdot 61 + \left(7 a + 46\right)\cdot 61^{2} + \left(41 a + 14\right)\cdot 61^{3} + \left(41 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 a + 25 + \left(26 a + 39\right)\cdot 61 + \left(5 a + 57\right)\cdot 61^{2} + \left(29 a + 34\right)\cdot 61^{3} + \left(34 a + 22\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 a + 48 + \left(41 a + 58\right)\cdot 61 + \left(53 a + 34\right)\cdot 61^{2} + \left(19 a + 48\right)\cdot 61^{3} + \left(19 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 42 a + 44 + \left(34 a + 46\right)\cdot 61 + \left(55 a + 36\right)\cdot 61^{2} + \left(31 a + 58\right)\cdot 61^{3} + \left(26 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 31 + 25\cdot 61 + 37\cdot 61^{2} + 40\cdot 61^{3} + 53\cdot 61^{4} +O\left(61^{ 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.