Properties

Label 2.3e2_283.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{2} \cdot 283 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2547= 3^{2} \cdot 283 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 6 x^{6} + 8 x^{5} + 42 x^{4} - 118 x^{3} + 95 x^{2} + 57 x - 119 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.283.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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