Properties

Label 2.643.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 643 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$643 $
Artin number field: Splitting field of $f=x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.643.2t1.1c1

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: \[ \begin{aligned} r_{ 1 } &= 369330905 a + 345500228 +O\left(37^{ 6 }\right) \\ r_{ 2 } &= -369330905 a + 986212408 +O\left(37^{ 6 }\right) \\ r_{ 3 } &= 591780527 a + 201913555 +O\left(37^{ 6 }\right) \\ r_{ 4 } &= -150761675 a + 608472593 +O\left(37^{ 6 }\right) \\ r_{ 5 } &= 1007590881 +O\left(37^{ 6 }\right) \\ r_{ 6 } &= -591780527 a + 1198967436 +O\left(37^{ 6 }\right) \\ r_{ 7 } &= 330640669 +O\left(37^{ 6 }\right) \\ r_{ 8 } &= 150761675 a + 452155050 +O\left(37^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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