Properties

Label 2.2e4_43.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{4} \cdot 43 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$688= 2^{4} \cdot 43 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{6} - 12 x^{5} + 16 x^{4} - 18 x^{3} + 10 x^{2} - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.43.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 7 a + 1 + \left(a + 15\right)\cdot 19 + \left(3 a + 4\right)\cdot 19^{2} + \left(2 a + 15\right)\cdot 19^{3} + 13 a\cdot 19^{4} + \left(6 a + 4\right)\cdot 19^{5} + \left(13 a + 13\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 8 + 10\cdot 19 + 5\cdot 19^{2} + 18\cdot 19^{3} + 16\cdot 19^{4} + 18\cdot 19^{5} +O\left(19^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 14 a + 16 + 5\cdot 19 + \left(2 a + 8\right)\cdot 19^{2} + \left(11 a + 5\right)\cdot 19^{3} + \left(3 a + 13\right)\cdot 19^{4} + \left(9 a + 6\right)\cdot 19^{5} + \left(2 a + 15\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 5 a + 11 + \left(18 a + 11\right)\cdot 19 + \left(16 a + 9\right)\cdot 19^{2} + \left(7 a + 14\right)\cdot 19^{3} + \left(15 a + 5\right)\cdot 19^{4} + \left(9 a + 12\right)\cdot 19^{5} + \left(16 a + 8\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 5 }$ $=$ $ a + 2 + \left(13 a + 15\right)\cdot 19 + \left(7 a + 14\right)\cdot 19^{2} + \left(2 a + 5\right)\cdot 19^{3} + \left(8 a + 3\right)\cdot 19^{4} + \left(12 a + 6\right)\cdot 19^{5} + \left(6 a + 11\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 12 a + 8 + \left(17 a + 9\right)\cdot 19 + \left(15 a + 6\right)\cdot 19^{2} + \left(16 a + 14\right)\cdot 19^{3} + \left(5 a + 11\right)\cdot 19^{4} + \left(12 a + 16\right)\cdot 19^{5} + 5 a\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 18 a + 3 + \left(5 a + 8\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + 16 a\cdot 19^{3} + \left(10 a + 9\right)\cdot 19^{4} + \left(6 a + 10\right)\cdot 19^{5} + \left(12 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 10 + 17\cdot 19^{2} + 19^{3} + 15\cdot 19^{4} + 19^{6} +O\left(19^{ 7 }\right)$

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

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

Character values on conjugacy classes

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