Properties

Label 2.2e3_43.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 43 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$344= 2^{3} \cdot 43 $
Artin number field: Splitting field of $f=x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 6 x^{3} - 2 x^{2} - 2$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.43.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= 484966176 a - 1231961661 +O\left(37^{ 6 }\right) \\ r_{ 2 } &= 816564238 a + 106783280 +O\left(37^{ 6 }\right) \\ r_{ 3 } &= 930307922 a + 163145809 +O\left(37^{ 6 }\right) \\ r_{ 4 } &= -113556636 +O\left(37^{ 6 }\right) \\ r_{ 5 } &= 1282221011 +O\left(37^{ 6 }\right) \\ r_{ 6 } &= -816564238 a - 1182572484 +O\left(37^{ 6 }\right) \\ r_{ 7 } &= -930307922 a + 251701291 +O\left(37^{ 6 }\right) \\ r_{ 8 } &= -484966176 a + 724239394 +O\left(37^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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