Properties

Label 2.2e3_11e2.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 11^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$968= 2^{3} \cdot 11^{2} $
Artin number field: Splitting field of $f=x^{8} + 22 x^{4} - 44 x^{3} - 44 x - 99$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.11.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= -364757456 a + 8464157 +O\left(13^{ 8 }\right) \\ r_{ 2 } &= 364757456 a + 306900024 +O\left(13^{ 8 }\right) \\ r_{ 3 } &= -10874964 a + 154513677 +O\left(13^{ 8 }\right) \\ r_{ 4 } &= 262005829 a + 104198988 +O\left(13^{ 8 }\right) \\ r_{ 5 } &= 129801865 +O\left(13^{ 8 }\right) \\ r_{ 6 } &= 10874964 a + 285013245 +O\left(13^{ 8 }\right) \\ r_{ 7 } &= -396213159 +O\left(13^{ 8 }\right) \\ r_{ 8 } &= -262005829 a + 223051924 +O\left(13^{ 8 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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