Properties

Label 2.163.8t12.1c2
Dimension 2
Group $\SL(2,3)$
Conductor $ 163 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\SL(2,3)$
Conductor:$163 $
Artin number field: Splitting field of $f=x^{8} - x^{7} + x^{6} - 4 x^{5} + 5 x^{4} - 8 x^{3} + 4 x^{2} - 8 x + 16$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 8T12
Parity: Even
Determinant: 1.163.3t1.1c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{3} + x + 40$
Roots: \[ \begin{aligned} r_{ 1 } &= 54114049 a^{2} - 25115108 a - 39218542 +O\left(43^{ 5 }\right) \\ r_{ 2 } &= -2097671 a^{2} + 63781279 a - 51373710 +O\left(43^{ 5 }\right) \\ r_{ 3 } &= -53911619 a^{2} - 30911808 a + 61092101 +O\left(43^{ 5 }\right) \\ r_{ 4 } &= 154124 +O\left(43^{ 5 }\right) \\ r_{ 5 } &= -65369940 +O\left(43^{ 5 }\right) \\ r_{ 6 } &= -44417676 a^{2} - 12835624 a - 6900730 +O\left(43^{ 5 }\right) \\ r_{ 7 } &= -9696373 a^{2} + 37950732 a - 32756009 +O\left(43^{ 5 }\right) \\ r_{ 8 } &= 56009290 a^{2} - 32869471 a - 12635736 +O\left(43^{ 5 }\right) \\ \end{aligned}\]

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,8)(2,7)(3,6)(4,5)$
$(1,5,2)(4,7,8)$
$(1,4,8,5)(2,3,7,6)$
$(1,3,8,6)(2,5,7,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$4$$3$$(1,5,2)(4,7,8)$$\zeta_{3} + 1$
$4$$3$$(1,2,5)(4,8,7)$$-\zeta_{3}$
$6$$4$$(1,4,8,5)(2,3,7,6)$$0$
$4$$6$$(1,4,2,8,5,7)(3,6)$$-\zeta_{3} - 1$
$4$$6$$(1,7,5,8,2,4)(3,6)$$\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.