Properties

Label 2.2e4_131.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{4} \cdot 131 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2096= 2^{4} \cdot 131 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} - 6 x^{6} + 28 x^{5} - 44 x^{4} + 38 x^{3} - 8 x^{2} - 4 x - 1 $ 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 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ a + 28 + \left(18 a + 1\right)\cdot 29 + \left(26 a + 18\right)\cdot 29^{2} + \left(22 a + 15\right)\cdot 29^{3} + \left(4 a + 8\right)\cdot 29^{4} + \left(a + 13\right)\cdot 29^{5} + \left(7 a + 26\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 9 + \left(19 a + 10\right)\cdot 29 + \left(28 a + 12\right)\cdot 29^{2} + \left(20 a + 15\right)\cdot 29^{3} + \left(9 a + 21\right)\cdot 29^{4} + \left(6 a + 2\right)\cdot 29^{5} + \left(18 a + 19\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 12 + 16\cdot 29 + 29^{2} + 13\cdot 29^{3} + 17\cdot 29^{4} + 22\cdot 29^{5} + 8\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 28 a + 4 + \left(10 a + 4\right)\cdot 29 + \left(2 a + 17\right)\cdot 29^{2} + \left(6 a + 16\right)\cdot 29^{3} + \left(24 a + 9\right)\cdot 29^{4} + \left(27 a + 14\right)\cdot 29^{5} + \left(21 a + 2\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 18 a + 11 + \left(13 a + 14\right)\cdot 29 + \left(16 a + 15\right)\cdot 29^{2} + \left(4 a + 2\right)\cdot 29^{3} + \left(22 a + 8\right)\cdot 29^{4} + \left(26 a + 11\right)\cdot 29^{5} + \left(9 a + 4\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 21 + 16\cdot 29 + 4\cdot 29^{2} + 10\cdot 29^{3} + 3\cdot 29^{4} + 20\cdot 29^{5} + 10\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 27 a + 19 + \left(9 a + 16\right)\cdot 29 + 20\cdot 29^{2} + \left(8 a + 4\right)\cdot 29^{3} + \left(19 a + 20\right)\cdot 29^{4} + \left(22 a + 24\right)\cdot 29^{5} + \left(10 a + 16\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 11 a + 14 + \left(15 a + 6\right)\cdot 29 + \left(12 a + 26\right)\cdot 29^{2} + \left(24 a + 8\right)\cdot 29^{3} + \left(6 a + 27\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} + \left(19 a + 27\right)\cdot 29^{6} +O\left(29^{ 7 }\right)$

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

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

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$$2$$(1,6)(3,5)(4,8)$$0$
$8$$3$$(1,6,7)(2,5,3)$$-1$
$6$$4$$(1,2,5,7)(3,4,6,8)$$0$
$8$$6$$(1,2,6,5,7,3)(4,8)$$1$
$6$$8$$(1,2,3,8,5,7,6,4)$$-\zeta_{8}^{3} - \zeta_{8}$
$6$$8$$(1,7,3,4,5,2,6,8)$$\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.