Properties

Label 10.3e14_17e8.30t176.4c1
Dimension 10
Group $S_6$
Conductor $ 3^{14} \cdot 17^{8}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$33364831591822329= 3^{14} \cdot 17^{8} $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + x^{3} + 14 x^{2} - 17 x + 6 $ 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 $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{2} + 192 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 67 + 139\cdot 193 + 134\cdot 193^{2} + 101\cdot 193^{3} + 74\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 98 + 189\cdot 193 + 100\cdot 193^{2} + 118\cdot 193^{3} + 21\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 176 a + 155 + \left(141 a + 115\right)\cdot 193 + \left(138 a + 43\right)\cdot 193^{2} + \left(172 a + 146\right)\cdot 193^{3} + \left(105 a + 112\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 98 a + 12 + \left(186 a + 175\right)\cdot 193 + \left(140 a + 55\right)\cdot 193^{2} + \left(98 a + 37\right)\cdot 193^{3} + \left(59 a + 85\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 95 a + 110 + \left(6 a + 70\right)\cdot 193 + \left(52 a + 10\right)\cdot 193^{2} + \left(94 a + 188\right)\cdot 193^{3} + \left(133 a + 45\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 17 a + 138 + \left(51 a + 81\right)\cdot 193 + \left(54 a + 40\right)\cdot 193^{2} + \left(20 a + 180\right)\cdot 193^{3} + \left(87 a + 45\right)\cdot 193^{4} +O\left(193^{ 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.