Properties

Label 10.3e14_31e6.30t88.2c1
Dimension 10
Group $A_6$
Conductor $ 3^{14} \cdot 31^{6}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$A_6$
Conductor:$4244902593608889= 3^{14} \cdot 31^{6} $
Artin number field: Splitting field of $f= x^{6} - x^{3} - 3 x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 95 + 81\cdot 101 + 30\cdot 101^{2} + 62\cdot 101^{3} + 50\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 66 a + 59 + \left(45 a + 55\right)\cdot 101 + \left(47 a + 81\right)\cdot 101^{2} + \left(83 a + 27\right)\cdot 101^{3} + \left(37 a + 55\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 a + 56 + \left(40 a + 89\right)\cdot 101 + \left(83 a + 13\right)\cdot 101^{2} + \left(26 a + 76\right)\cdot 101^{3} + 41\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 82 a + 31 + \left(60 a + 29\right)\cdot 101 + \left(17 a + 4\right)\cdot 101^{2} + \left(74 a + 100\right)\cdot 101^{3} + \left(100 a + 15\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 42 + 76\cdot 101 + 47\cdot 101^{2} + 25\cdot 101^{3} + 16\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 35 a + 20 + \left(55 a + 71\right)\cdot 101 + \left(53 a + 23\right)\cdot 101^{2} + \left(17 a + 11\right)\cdot 101^{3} + \left(63 a + 22\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$10$
$45$$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)$$0$
$72$$5$$(1,2,3,4,5)$$0$
$72$$5$$(1,3,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.