Properties

Label 10.2e18_67e6.30t88.1c1
Dimension 10
Group $A_6$
Conductor $ 2^{18} \cdot 67^{6}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$A_6$
Conductor:$23713122135310336= 2^{18} \cdot 67^{6} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{4} + x^{2} - 2 x - 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 $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 28 a + 5 + \left(26 a + 29\right)\cdot 31 + \left(6 a + 1\right)\cdot 31^{2} + \left(20 a + 13\right)\cdot 31^{3} + \left(12 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 28\cdot 31 + 22\cdot 31^{2} + 15\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 a + 26 + \left(18 a + 4\right)\cdot 31 + \left(a + 22\right)\cdot 31^{2} + \left(6 a + 21\right)\cdot 31^{3} + \left(4 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 30 + \left(4 a + 23\right)\cdot 31 + \left(24 a + 19\right)\cdot 31^{2} + \left(10 a + 15\right)\cdot 31^{3} + \left(18 a + 9\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 13 + \left(12 a + 1\right)\cdot 31 + \left(29 a + 7\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(26 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 2 + 5\cdot 31 + 19\cdot 31^{2} + 25\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 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.