Properties

Label 5.6718464.6t15.b.a
Dimension $5$
Group $A_6$
Conductor $6718464$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $A_6$
Conductor: \(6718464\)\(\medspace = 2^{10} \cdot 3^{8} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.6718464.1
Galois orbit size: $1$
Smallest permutation container: $A_6$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_6$
Projective stem field: Galois closure of 6.2.6718464.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{4} - 12x^{3} - 9x^{2} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{2} + 63x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 58 a + 3 + \left(38 a + 63\right)\cdot 67 + \left(5 a + 26\right)\cdot 67^{2} + \left(47 a + 2\right)\cdot 67^{3} + \left(42 a + 54\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a + 34 + \left(28 a + 26\right)\cdot 67 + \left(61 a + 10\right)\cdot 67^{2} + \left(19 a + 51\right)\cdot 67^{3} + \left(24 a + 43\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 63 + 33\cdot 67 + 21\cdot 67^{2} + 54\cdot 67^{3} + 2\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 51 + 31\cdot 67 + 9\cdot 67^{2} + 27\cdot 67^{3} + 41\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 59 a + 41 + \left(45 a + 27\right)\cdot 67 + \left(54 a + 13\right)\cdot 67^{2} + \left(34 a + 24\right)\cdot 67^{3} + \left(53 a + 40\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + 9 + \left(21 a + 18\right)\cdot 67 + \left(12 a + 52\right)\cdot 67^{2} + \left(32 a + 41\right)\cdot 67^{3} + \left(13 a + 18\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display

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$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$72$$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.