Properties

Label 2.1399.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 1399 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1399 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 4 x^{4} + 4 x^{3} - 14 x^{2} + 11 x - 8 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.1399.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 31 + 21\cdot 47 + 11\cdot 47^{2} + 22\cdot 47^{3} + 43\cdot 47^{4} + 39\cdot 47^{5} + 11\cdot 47^{6} + 23\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 28 a + 35 + 4 a\cdot 47 + \left(6 a + 8\right)\cdot 47^{2} + \left(38 a + 6\right)\cdot 47^{3} + \left(2 a + 26\right)\cdot 47^{4} + \left(19 a + 30\right)\cdot 47^{5} + \left(34 a + 40\right)\cdot 47^{6} + \left(28 a + 1\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 17 + 25\cdot 47 + 35\cdot 47^{2} + 24\cdot 47^{3} + 3\cdot 47^{4} + 7\cdot 47^{5} + 35\cdot 47^{6} + 23\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 22 + \left(26 a + 45\right)\cdot 47 + \left(13 a + 22\right)\cdot 47^{2} + \left(42 a + 11\right)\cdot 47^{3} + 20\cdot 47^{4} + \left(41 a + 6\right)\cdot 47^{5} + \left(14 a + 29\right)\cdot 47^{6} + \left(42 a + 35\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 28 a + 4 + \left(4 a + 18\right)\cdot 47 + \left(6 a + 31\right)\cdot 47^{2} + \left(38 a + 17\right)\cdot 47^{3} + \left(2 a + 6\right)\cdot 47^{4} + \left(19 a + 28\right)\cdot 47^{5} + \left(34 a + 3\right)\cdot 47^{6} + \left(28 a + 22\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 19 a + 13 + \left(42 a + 46\right)\cdot 47 + \left(40 a + 38\right)\cdot 47^{2} + \left(8 a + 40\right)\cdot 47^{3} + \left(44 a + 20\right)\cdot 47^{4} + \left(27 a + 16\right)\cdot 47^{5} + \left(12 a + 6\right)\cdot 47^{6} + \left(18 a + 45\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 45 a + 26 + \left(20 a + 1\right)\cdot 47 + \left(33 a + 24\right)\cdot 47^{2} + \left(4 a + 35\right)\cdot 47^{3} + \left(46 a + 26\right)\cdot 47^{4} + \left(5 a + 40\right)\cdot 47^{5} + \left(32 a + 17\right)\cdot 47^{6} + \left(4 a + 11\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 19 a + 44 + \left(42 a + 28\right)\cdot 47 + \left(40 a + 15\right)\cdot 47^{2} + \left(8 a + 29\right)\cdot 47^{3} + \left(44 a + 40\right)\cdot 47^{4} + \left(27 a + 18\right)\cdot 47^{5} + \left(12 a + 43\right)\cdot 47^{6} + \left(18 a + 24\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$

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

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

Character values on conjugacy classes

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