Properties

Label 2.1823.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 1823 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1823 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 7 x^{6} - 6 x^{5} - 4 x^{4} - 3 x^{3} + 6 x^{2} - 5 x - 19 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.1823.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 26 + 7\cdot 31 + 27\cdot 31^{2} + 24\cdot 31^{4} + 22\cdot 31^{5} + 9\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 10 a + 6 + \left(8 a + 29\right)\cdot 31 + 30\cdot 31^{2} + \left(19 a + 8\right)\cdot 31^{3} + \left(27 a + 18\right)\cdot 31^{4} + \left(11 a + 1\right)\cdot 31^{5} + \left(28 a + 27\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 21 a + 26 + \left(22 a + 4\right)\cdot 31 + \left(30 a + 23\right)\cdot 31^{2} + \left(11 a + 15\right)\cdot 31^{3} + \left(3 a + 23\right)\cdot 31^{4} + \left(19 a + 28\right)\cdot 31^{5} + \left(2 a + 9\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 25 + \left(23 a + 1\right)\cdot 31 + \left(24 a + 4\right)\cdot 31^{2} + \left(2 a + 8\right)\cdot 31^{3} + \left(4 a + 10\right)\cdot 31^{4} + \left(10 a + 4\right)\cdot 31^{5} + \left(a + 8\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 4 + \left(21 a + 26\right)\cdot 31 + \left(9 a + 30\right)\cdot 31^{2} + \left(7 a + 12\right)\cdot 31^{3} + \left(30 a + 7\right)\cdot 31^{4} + \left(7 a + 16\right)\cdot 31^{5} + \left(6 a + 16\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 2 + 5\cdot 31 + 10\cdot 31^{2} + 8\cdot 31^{3} + 26\cdot 31^{4} + 27\cdot 31^{5} + 30\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 28 a + 10 + \left(9 a + 3\right)\cdot 31 + \left(21 a + 29\right)\cdot 31^{2} + \left(23 a + 17\right)\cdot 31^{3} + 29\cdot 31^{4} + \left(23 a + 1\right)\cdot 31^{5} + \left(24 a + 21\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 29 a + 29 + \left(7 a + 14\right)\cdot 31 + \left(6 a + 30\right)\cdot 31^{2} + \left(28 a + 19\right)\cdot 31^{3} + \left(26 a + 15\right)\cdot 31^{4} + \left(20 a + 20\right)\cdot 31^{5} + 29 a\cdot 31^{6} +O\left(31^{ 7 }\right)$

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

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

Character values on conjugacy classes

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