Properties

Label 10.3e12_461e4.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 3^{12} \cdot 461^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$24002626001540481= 3^{12} \cdot 461^{4} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 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 $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 97 a + 77 + \left(40 a + 81\right)\cdot 127 + \left(86 a + 81\right)\cdot 127^{2} + \left(123 a + 68\right)\cdot 127^{3} + \left(4 a + 66\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 114 a + 53 + \left(45 a + 117\right)\cdot 127 + \left(66 a + 125\right)\cdot 127^{2} + \left(5 a + 93\right)\cdot 127^{3} + \left(20 a + 16\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 a + 71 + \left(27 a + 52\right)\cdot 127 + \left(120 a + 30\right)\cdot 127^{2} + \left(68 a + 65\right)\cdot 127^{3} + \left(74 a + 29\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 a + 40 + \left(81 a + 49\right)\cdot 127 + \left(60 a + 19\right)\cdot 127^{2} + \left(121 a + 33\right)\cdot 127^{3} + \left(106 a + 31\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 102 a + 96 + \left(99 a + 54\right)\cdot 127 + \left(6 a + 123\right)\cdot 127^{2} + \left(58 a + 13\right)\cdot 127^{3} + \left(52 a + 35\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 30 a + 47 + \left(86 a + 25\right)\cdot 127 + 40 a\cdot 127^{2} + \left(3 a + 106\right)\cdot 127^{3} + \left(122 a + 74\right)\cdot 127^{4} +O\left(127^{ 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.