Properties

Label 10.3e6_167e6.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 3^{6} \cdot 167^{6}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$15813440003753001= 3^{6} \cdot 167^{6} $
Artin number field: Splitting field of $f= x^{6} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 36 a + 14 + \left(6 a + 36\right)\cdot 43 + \left(23 a + 4\right)\cdot 43^{2} + \left(34 a + 40\right)\cdot 43^{3} + \left(6 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 + 25\cdot 43 + 37\cdot 43^{2} + 12\cdot 43^{3} + 19\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 7 + \left(36 a + 7\right)\cdot 43 + \left(19 a + 21\right)\cdot 43^{2} + \left(8 a + 8\right)\cdot 43^{3} + \left(36 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 3 + \left(6 a + 25\right)\cdot 43 + \left(8 a + 34\right)\cdot 43^{2} + \left(41 a + 9\right)\cdot 43^{3} + \left(22 a + 14\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 + 9\cdot 43 + 37\cdot 43^{2} + 14\cdot 43^{3} + 9\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 a + 9 + \left(36 a + 25\right)\cdot 43 + \left(34 a + 36\right)\cdot 43^{2} + \left(a + 42\right)\cdot 43^{3} + \left(20 a + 38\right)\cdot 43^{4} +O\left(43^{ 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$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$2$
$15$$2$$(1,2)$$-2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$1$
The blue line marks the conjugacy class containing complex conjugation.