Properties

Label 2.751.24t22.2c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 751 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$751 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 20 x^{3} - 14 x^{2} + 13 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:
$r_{ 1 }$ $=$ $ 2 a + \left(22 a + 20\right)\cdot 29 + \left(19 a + 12\right)\cdot 29^{2} + \left(24 a + 27\right)\cdot 29^{3} + \left(7 a + 24\right)\cdot 29^{4} + \left(15 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 27 a + 10 + \left(6 a + 12\right)\cdot 29 + \left(9 a + 2\right)\cdot 29^{2} + \left(4 a + 15\right)\cdot 29^{3} + \left(21 a + 10\right)\cdot 29^{4} + 13 a\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 4 + \left(16 a + 16\right)\cdot 29 + \left(11 a + 23\right)\cdot 29^{2} + \left(23 a + 22\right)\cdot 29^{3} + \left(7 a + 14\right)\cdot 29^{4} + \left(20 a + 6\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 28 a + 17 + \left(7 a + 28\right)\cdot 29 + 24\cdot 29^{2} + \left(12 a + 23\right)\cdot 29^{3} + \left(19 a + 25\right)\cdot 29^{4} + a\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 10 + \left(12 a + 3\right)\cdot 29 + \left(17 a + 7\right)\cdot 29^{2} + \left(5 a + 12\right)\cdot 29^{3} + \left(21 a + 1\right)\cdot 29^{4} + \left(8 a + 13\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 13 + 29 + 23\cdot 29^{2} + 7\cdot 29^{3} + 2\cdot 29^{4} + 5\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 23 + 22\cdot 29 + 3\cdot 29^{2} + 10\cdot 29^{3} + 12\cdot 29^{4} + 23\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 8 }$ $=$ $ a + 12 + \left(21 a + 11\right)\cdot 29 + \left(28 a + 18\right)\cdot 29^{2} + \left(16 a + 25\right)\cdot 29^{3} + \left(9 a + 23\right)\cdot 29^{4} + \left(27 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

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

Character values on conjugacy classes

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