Properties

Label 8.388...016.36t555.b.b
Dimension $8$
Group $A_6$
Conductor $3.881\times 10^{13}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $A_6$
Conductor: \(38806720086016\)\(\medspace = 2^{18} \cdot 23^{6}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.71639296.1
Galois orbit size: $2$
Smallest permutation container: $A_6$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_6$
Projective field: Galois closure of 6.2.71639296.1

Defining polynomial

$f(x)$$=$$ x^{6} - x^{5} + x^{4} + 4 x^{3} - 6 x^{2} + 2 x + 2 $.

The roots of $f$ are computed in $\Q_{ 641 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 210 + 349\cdot 641 + 367\cdot 641^{2} + 225\cdot 641^{3} + 591\cdot 641^{4} +O\left(641^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 292 + 506\cdot 641 + 51\cdot 641^{2} + 40\cdot 641^{3} + 75\cdot 641^{4} +O\left(641^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 367 + 327\cdot 641 + 583\cdot 641^{2} + 387\cdot 641^{3} + 364\cdot 641^{4} +O\left(641^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 493 + 420\cdot 641 + 417\cdot 641^{2} + 521\cdot 641^{3} + 72\cdot 641^{4} +O\left(641^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 599 + 165\cdot 641 + 550\cdot 641^{2} + 558\cdot 641^{3} + 188\cdot 641^{4} +O\left(641^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 604 + 152\cdot 641 + 593\cdot 641^{2} + 188\cdot 641^{3} + 630\cdot 641^{4} +O\left(641^{ 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$$1$$()$$8$
$45$$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}$
$72$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.