Properties

Label 2.751.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 751 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$751 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 2 x^{6} + 4 x^{5} - 9 x^{4} + 17 x^{3} - 21 x^{2} + 7 x - 4 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.751.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 9 a + 12 + \left(a + 10\right)\cdot 29 + \left(22 a + 19\right)\cdot 29^{2} + \left(13 a + 6\right)\cdot 29^{3} + \left(7 a + 3\right)\cdot 29^{4} + \left(a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 13 + \left(27 a + 26\right)\cdot 29 + \left(19 a + 6\right)\cdot 29^{2} + \left(19 a + 27\right)\cdot 29^{3} + \left(22 a + 1\right)\cdot 29^{4} + 9\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 16 + 28\cdot 29 + 22\cdot 29^{2} + 16\cdot 29^{3} + 12\cdot 29^{4} + 11\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 18 a + 10 + \left(a + 7\right)\cdot 29 + \left(9 a + 21\right)\cdot 29^{2} + \left(9 a + 18\right)\cdot 29^{3} + \left(6 a + 8\right)\cdot 29^{4} + \left(28 a + 19\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 13 + 23\cdot 29 + \left(7 a + 21\right)\cdot 29^{2} + 11\cdot 29^{3} + \left(10 a + 28\right)\cdot 29^{4} + \left(5 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 18 + 6\cdot 29 + 13\cdot 29^{2} + 4\cdot 29^{3} + 14\cdot 29^{4} + 27\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 20 a + 28 + \left(27 a + 7\right)\cdot 29 + \left(6 a + 12\right)\cdot 29^{2} + \left(15 a + 24\right)\cdot 29^{3} + \left(21 a + 26\right)\cdot 29^{4} + \left(27 a + 25\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 7 a + 7 + \left(28 a + 5\right)\cdot 29 + \left(21 a + 27\right)\cdot 29^{2} + \left(28 a + 5\right)\cdot 29^{3} + \left(18 a + 20\right)\cdot 29^{4} + \left(23 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

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

Character values on conjugacy classes

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