Properties

Label 9.2e18_3e16.10t26.1c1
Dimension 9
Group $A_6$
Conductor $ 2^{18} \cdot 3^{16}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$9$
Group:$A_6$
Conductor:$11284439629824= 2^{18} \cdot 3^{16} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PSL(2,9)$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 65 a + 49 + \left(29 a + 24\right)\cdot 67 + \left(44 a + 10\right)\cdot 67^{2} + \left(25 a + 60\right)\cdot 67^{3} + \left(31 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 41 + \left(37 a + 12\right)\cdot 67 + \left(22 a + 24\right)\cdot 67^{2} + \left(41 a + 51\right)\cdot 67^{3} + \left(35 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 58\cdot 67 + 31\cdot 67^{2} + 61\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 + 66\cdot 67 + 45\cdot 67^{2} + 42\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 45 + \left(18 a + 52\right)\cdot 67 + \left(36 a + 47\right)\cdot 67^{2} + \left(16 a + 44\right)\cdot 67^{3} + \left(58 a + 1\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 62 a + 65 + \left(48 a + 52\right)\cdot 67 + \left(30 a + 40\right)\cdot 67^{2} + \left(50 a + 7\right)\cdot 67^{3} + \left(8 a + 17\right)\cdot 67^{4} +O\left(67^{ 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$$1$$()$$9$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$0$
$40$$3$$(1,2,3)$$0$
$90$$4$$(1,2,3,4)(5,6)$$1$
$72$$5$$(1,2,3,4,5)$$-1$
$72$$5$$(1,3,4,5,2)$$-1$
The blue line marks the conjugacy class containing complex conjugation.