Properties

Label 8.604...881.36t555.a.a
Dimension $8$
Group $A_6$
Conductor $6.044\times 10^{19}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $A_6$
Conductor: \(60437550349593857881\)\(\medspace = 7^{6} \cdot 283^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.6.192293689.1
Galois orbit size: $2$
Smallest permutation container: $A_6$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_6$
Projective stem field: Galois closure of 6.6.192293689.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 12x^{4} + 7x^{3} + 21x^{2} - 7x - 7 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 32 + 52\cdot 53 + 38\cdot 53^{2} + 19\cdot 53^{3} + 9\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 40 a + 5 + \left(5 a + 34\right)\cdot 53 + \left(3 a + 52\right)\cdot 53^{2} + \left(49 a + 16\right)\cdot 53^{3} + \left(4 a + 28\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 + 37\cdot 53 + 36\cdot 53^{2} + 24\cdot 53^{3} + 13\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + 6 + \left(47 a + 17\right)\cdot 53 + \left(49 a + 6\right)\cdot 53^{2} + \left(3 a + 51\right)\cdot 53^{3} + \left(48 a + 51\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a + 35 + \left(34 a + 30\right)\cdot 53 + \left(28 a + 51\right)\cdot 53^{2} + \left(36 a + 43\right)\cdot 53^{3} + \left(50 a + 50\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 31 a + 17 + \left(18 a + 40\right)\cdot 53 + \left(24 a + 25\right)\cdot 53^{2} + \left(16 a + 2\right)\cdot 53^{3} + \left(2 a + 5\right)\cdot 53^{4} +O(53^{5})\) 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$$1$$()$$8$
$45$$2$$(1,2)(3,4)$$0$
$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$$(1,2,3,4,5)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$72$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.