Properties

Label 2.643.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 643 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$643 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.643.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 13 a + \left(19 a + 6\right)\cdot 37 + \left(14 a + 34\right)\cdot 37^{2} + \left(2 a + 12\right)\cdot 37^{3} + \left(12 a + 36\right)\cdot 37^{4} + \left(5 a + 4\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 24 a + 15 + \left(17 a + 33\right)\cdot 37 + \left(22 a + 35\right)\cdot 37^{2} + \left(34 a + 7\right)\cdot 37^{3} + \left(24 a + 8\right)\cdot 37^{4} + \left(31 a + 14\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 4 + \left(4 a + 30\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(28 a + 27\right)\cdot 37^{3} + \left(19 a + 33\right)\cdot 37^{4} + \left(8 a + 2\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 5 a + 8 + 22 a\cdot 37 + \left(23 a + 21\right)\cdot 37^{2} + \left(20 a + 24\right)\cdot 37^{3} + \left(30 a + 28\right)\cdot 37^{4} + \left(34 a + 8\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 36 + 37^{2} + 23\cdot 37^{3} + 19\cdot 37^{4} + 14\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 26 a + 11 + \left(32 a + 36\right)\cdot 37 + \left(35 a + 7\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(17 a + 10\right)\cdot 37^{4} + \left(28 a + 17\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 11 + 31\cdot 37 + 20\cdot 37^{2} + 15\cdot 37^{3} + 28\cdot 37^{4} + 4\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 32 a + 28 + \left(14 a + 9\right)\cdot 37 + \left(13 a + 19\right)\cdot 37^{2} + \left(16 a + 9\right)\cdot 37^{3} + \left(6 a + 19\right)\cdot 37^{4} + \left(2 a + 6\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$

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

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

Character values on conjugacy classes

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