Properties

Label 2.331.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 331 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$331 $
Artin number field: Splitting field of $f=x^{8} - 4 x^{7} + 5 x^{6} - x^{5} - 3 x^{4} + 3 x^{3} - x - 1$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.331.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= -413031076564 a + 210360019362 +O\left(13^{ 11 }\right) \\ r_{ 2 } &= 319848645846 a - 769148715979 +O\left(13^{ 11 }\right) \\ r_{ 3 } &= 413031076564 a - 209748243981 +O\left(13^{ 11 }\right) \\ r_{ 4 } &= -413031076564 a + 209748243982 +O\left(13^{ 11 }\right) \\ r_{ 5 } &= 328864739667 +O\left(13^{ 11 }\right) \\ r_{ 6 } &= 413031076564 a - 210360019361 +O\left(13^{ 11 }\right) \\ r_{ 7 } &= -328864739666 +O\left(13^{ 11 }\right) \\ r_{ 8 } &= -319848645846 a + 769148715980 +O\left(13^{ 11 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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