Properties

Label 2.1816.24t22.a
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 227 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1816= 2^{3} \cdot 227 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} - 4 x^{5} - 6 x^{4} - 10 x^{3} + 6 x^{2} + 4 x - 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Projective image: $S_4$
Projective field: Galois closure of 4.2.3632.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,5)(2,7)(3,8)(4,6)$ $-2$ $-2$
$12$ $2$ $(2,3)(4,6)(7,8)$ $0$ $0$
$8$ $3$ $(1,4,7)(2,5,6)$ $-1$ $-1$
$6$ $4$ $(1,2,5,7)(3,6,8,4)$ $0$ $0$
$8$ $6$ $(1,3,6,5,8,4)(2,7)$ $1$ $1$
$6$ $8$ $(1,2,6,3,5,7,4,8)$ $-\zeta_{8}^{3} - \zeta_{8}$ $\zeta_{8}^{3} + \zeta_{8}$
$6$ $8$ $(1,7,6,8,5,2,4,3)$ $\zeta_{8}^{3} + \zeta_{8}$ $-\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.