Properties

Label 2.2e4_43.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{4} \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:$688= 2^{4} \cdot 43 $
Artin number field: Splitting field of $f=x^{8} - 2 x^{7} + 4 x^{6} - 12 x^{5} + 16 x^{4} - 18 x^{3} + 10 x^{2} - 1$ 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 $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= -265709692 a - 272266275 +O\left(19^{ 7 }\right) \\ r_{ 2 } &= 46780383 +O\left(19^{ 7 }\right) \\ r_{ 3 } &= 116843801 a - 171595463 +O\left(19^{ 7 }\right) \\ r_{ 4 } &= -116843801 a + 406831336 +O\left(19^{ 7 }\right) \\ r_{ 5 } &= 313047535 a - 361079855 +O\left(19^{ 7 }\right) \\ r_{ 6 } &= 265709692 a + 41149486 +O\left(19^{ 7 }\right) \\ r_{ 7 } &= -313047535 a + 261166688 +O\left(19^{ 7 }\right) \\ r_{ 8 } &= 49013702 +O\left(19^{ 7 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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