Properties

Label 10.17e6_101e4.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 17^{6} \cdot 101^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$2511765109305169= 17^{6} \cdot 101^{4} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} + x^{3} - 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 $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 97 a + 59 + \left(40 a + 17\right)\cdot 109 + \left(86 a + 40\right)\cdot 109^{2} + \left(2 a + 24\right)\cdot 109^{3} + \left(15 a + 40\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 80 + \left(a + 76\right)\cdot 109 + \left(92 a + 36\right)\cdot 109^{2} + \left(3 a + 18\right)\cdot 109^{3} + \left(82 a + 75\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 47 + \left(68 a + 70\right)\cdot 109 + \left(22 a + 85\right)\cdot 109^{2} + \left(106 a + 49\right)\cdot 109^{3} + \left(93 a + 52\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 93 a + 37 + \left(33 a + 74\right)\cdot 109 + \left(3 a + 33\right)\cdot 109^{2} + \left(82 a + 58\right)\cdot 109^{3} + \left(51 a + 72\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 104 a + 85 + \left(107 a + 72\right)\cdot 109 + \left(16 a + 18\right)\cdot 109^{2} + \left(105 a + 39\right)\cdot 109^{3} + \left(26 a + 44\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 21 + \left(75 a + 15\right)\cdot 109 + \left(105 a + 3\right)\cdot 109^{2} + \left(26 a + 28\right)\cdot 109^{3} + \left(57 a + 42\right)\cdot 109^{4} +O\left(109^{ 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.