Properties

Label 2.2e3_131.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 131 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1048= 2^{3} \cdot 131 $
Artin number field: Splitting field of $f= x^{8} + 4 x^{6} - 4 x^{5} + 2 x^{4} - 6 x^{3} - 4 x^{2} - 18 x - 29 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.131.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 8\cdot 29 + 27\cdot 29^{3} + 22\cdot 29^{4} + 2\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 1 + \left(24 a + 19\right)\cdot 29 + 4\cdot 29^{2} + \left(10 a + 3\right)\cdot 29^{3} + \left(20 a + 1\right)\cdot 29^{4} + \left(6 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 24 a + \left(7 a + 14\right)\cdot 29 + \left(9 a + 9\right)\cdot 29^{2} + \left(3 a + 23\right)\cdot 29^{3} + \left(21 a + 26\right)\cdot 29^{4} + \left(26 a + 9\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 5 a + 4 + 21 a\cdot 29 + \left(19 a + 19\right)\cdot 29^{2} + \left(25 a + 1\right)\cdot 29^{3} + \left(7 a + 13\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 7 + 22\cdot 29 + 18\cdot 29^{2} + 19\cdot 29^{3} + 7\cdot 29^{4} + 25\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 28 a + 3 + \left(3 a + 20\right)\cdot 29 + \left(4 a + 16\right)\cdot 29^{2} + \left(2 a + 5\right)\cdot 29^{3} + \left(17 a + 21\right)\cdot 29^{4} + \left(23 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 26 a + 16 + \left(4 a + 20\right)\cdot 29 + \left(28 a + 13\right)\cdot 29^{2} + \left(18 a + 23\right)\cdot 29^{3} + \left(8 a + 5\right)\cdot 29^{4} + \left(22 a + 4\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 8 }$ $=$ $ a + 27 + \left(25 a + 11\right)\cdot 29 + \left(24 a + 4\right)\cdot 29^{2} + \left(26 a + 12\right)\cdot 29^{3} + \left(11 a + 17\right)\cdot 29^{4} + \left(5 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

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

Character values on conjugacy classes

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