Properties

Label 2.2e2_571.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{2} \cdot 571 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2284= 2^{2} \cdot 571 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 9 x^{6} - 13 x^{5} + 11 x^{4} - 5 x^{3} + 2 x^{2} - x - 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.571.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 8 a + 22 + \left(21 a + 14\right)\cdot 41 + \left(3 a + 5\right)\cdot 41^{2} + \left(2 a + 21\right)\cdot 41^{3} + \left(40 a + 31\right)\cdot 41^{4} + \left(7 a + 32\right)\cdot 41^{5} + \left(a + 5\right)\cdot 41^{6} + \left(28 a + 33\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 33 a + 20 + \left(19 a + 26\right)\cdot 41 + \left(37 a + 35\right)\cdot 41^{2} + \left(38 a + 19\right)\cdot 41^{3} + 9\cdot 41^{4} + \left(33 a + 8\right)\cdot 41^{5} + \left(39 a + 35\right)\cdot 41^{6} + \left(12 a + 7\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 37 + \left(21 a + 11\right)\cdot 41 + \left(3 a + 5\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(40 a + 14\right)\cdot 41^{4} + \left(7 a + 24\right)\cdot 41^{5} + \left(a + 39\right)\cdot 41^{6} + \left(28 a + 6\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 15 a + 19 + \left(26 a + 29\right)\cdot 41 + \left(23 a + 18\right)\cdot 41^{2} + \left(29 a + 8\right)\cdot 41^{3} + \left(12 a + 16\right)\cdot 41^{4} + \left(6 a + 17\right)\cdot 41^{5} + \left(25 a + 6\right)\cdot 41^{6} + \left(10 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 5 + 32\cdot 41 + 18\cdot 41^{2} + 17\cdot 41^{3} + 8\cdot 41^{4} + 2\cdot 41^{5} + 28\cdot 41^{6} + 33\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 33 a + 5 + \left(19 a + 29\right)\cdot 41 + \left(37 a + 35\right)\cdot 41^{2} + \left(38 a + 23\right)\cdot 41^{3} + 26\cdot 41^{4} + \left(33 a + 16\right)\cdot 41^{5} + \left(39 a + 1\right)\cdot 41^{6} + \left(12 a + 34\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 37 + 8\cdot 41 + 22\cdot 41^{2} + 23\cdot 41^{3} + 32\cdot 41^{4} + 38\cdot 41^{5} + 12\cdot 41^{6} + 7\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 26 a + 23 + \left(14 a + 11\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(11 a + 32\right)\cdot 41^{3} + \left(28 a + 24\right)\cdot 41^{4} + \left(34 a + 23\right)\cdot 41^{5} + \left(15 a + 34\right)\cdot 41^{6} + \left(30 a + 23\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$

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

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

Character values on conjugacy classes

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