Properties

Label 10.2e24_223e4.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 2^{24} \cdot 223^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$41489609581920256= 2^{24} \cdot 223^{4} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 4 x^{3} - 4 x^{2} + 2 $ 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 $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{2} + 190 x + 19 $
Roots:
$r_{ 1 }$ $=$ $ 80 + 69\cdot 191 + 27\cdot 191^{2} + 131\cdot 191^{3} + 20\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 62 a + 39 + \left(113 a + 96\right)\cdot 191 + \left(93 a + 126\right)\cdot 191^{2} + \left(108 a + 17\right)\cdot 191^{3} + \left(80 a + 158\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 45 + 174\cdot 191 + 131\cdot 191^{2} + 2\cdot 191^{3} + 176\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 54 + \left(71 a + 108\right)\cdot 191 + \left(65 a + 188\right)\cdot 191^{2} + \left(5 a + 128\right)\cdot 191^{3} + \left(59 a + 112\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 180 a + 65 + \left(119 a + 168\right)\cdot 191 + \left(125 a + 182\right)\cdot 191^{2} + \left(185 a + 68\right)\cdot 191^{3} + \left(131 a + 166\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 129 a + 101 + \left(77 a + 147\right)\cdot 191 + \left(97 a + 106\right)\cdot 191^{2} + \left(82 a + 32\right)\cdot 191^{3} + \left(110 a + 130\right)\cdot 191^{4} +O\left(191^{ 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.