Properties

Label 2.751.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 751 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$751 $
Artin number field: Splitting field of $f=x^{8} - x^{7} - 2 x^{6} + 4 x^{5} - 9 x^{4} + 17 x^{3} - 21 x^{2} + 7 x - 4$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.751.2t1.1c1

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: \[ \begin{aligned} r_{ 1 } &= 25797713 a - 38737840 +O\left(29^{ 6 }\right) \\ r_{ 2 } &= 16040346 a + 185971938 +O\left(29^{ 6 }\right) \\ r_{ 3 } &= 234519565 +O\left(29^{ 6 }\right) \\ r_{ 4 } &= -16040346 a - 198996366 +O\left(29^{ 6 }\right) \\ r_{ 5 } &= 109634464 a + 81623935 +O\left(29^{ 6 }\right) \\ r_{ 6 } &= -31011683 +O\left(29^{ 6 }\right) \\ r_{ 7 } &= -25797713 a - 63059631 +O\left(29^{ 6 }\right) \\ r_{ 8 } &= -109634464 a - 170309917 +O\left(29^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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