Properties

Label 2.3e2_109.24t22.3c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{2} \cdot 109 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$981= 3^{2} \cdot 109 $
Artin number field: Splitting field of $f=x^{8} - 3 x^{7} + 6 x^{6} - 6 x^{5} + 12 x^{3} - 18 x^{2} + 12 x - 3$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3_109.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= 177132452 a - 110794950 +O\left(29^{ 6 }\right) \\ r_{ 2 } &= -197390469 a - 53190316 +O\left(29^{ 6 }\right) \\ r_{ 3 } &= 124806690 +O\left(29^{ 6 }\right) \\ r_{ 4 } &= 197390469 a - 74405628 +O\left(29^{ 6 }\right) \\ r_{ 5 } &= 33783805 a + 128475842 +O\left(29^{ 6 }\right) \\ r_{ 6 } &= -177132452 a - 198210551 +O\left(29^{ 6 }\right) \\ r_{ 7 } &= 270831073 +O\left(29^{ 6 }\right) \\ r_{ 8 } &= -33783805 a - 87512157 +O\left(29^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,6)(3,7)(5,8)$$-2$
$12$$2$$(1,4)(2,8)(5,6)$$0$
$8$$3$$(1,7,8)(3,5,4)$$-1$
$6$$4$$(1,7,4,3)(2,5,6,8)$$0$
$8$$6$$(1,4)(2,5,3,6,8,7)$$1$
$6$$8$$(1,8,3,6,4,5,7,2)$$\zeta_{8}^{3} + \zeta_{8}$
$6$$8$$(1,5,3,2,4,8,7,6)$$-\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.