Properties

Label 2.1423.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 1423 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1423 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + x^{6} + 7 x^{5} - 13 x^{4} - 4 x^{3} + 23 x^{2} - 25 x - 4 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.1423.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 27 a + 25 + \left(15 a + 13\right)\cdot 31 + \left(15 a + 14\right)\cdot 31^{2} + \left(21 a + 10\right)\cdot 31^{3} + \left(29 a + 25\right)\cdot 31^{4} + \left(18 a + 16\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 10 a + 24 + \left(9 a + 20\right)\cdot 31 + \left(7 a + 27\right)\cdot 31^{2} + \left(12 a + 29\right)\cdot 31^{3} + \left(8 a + 18\right)\cdot 31^{4} + \left(14 a + 1\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 26 + 26\cdot 31 + 2\cdot 31^{2} + 27\cdot 31^{3} + 22\cdot 31^{4} + 27\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 8 a + 12 + 7\cdot 31 + \left(3 a + 29\right)\cdot 31^{2} + \left(18 a + 11\right)\cdot 31^{3} + \left(19 a + 7\right)\cdot 31^{4} + \left(15 a + 10\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 4 a + 17 + \left(15 a + 18\right)\cdot 31 + \left(15 a + 29\right)\cdot 31^{2} + \left(9 a + 6\right)\cdot 31^{3} + \left(a + 1\right)\cdot 31^{4} + \left(12 a + 25\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 12 + 7\cdot 31 + 14\cdot 31^{2} + 7\cdot 31^{3} + 27\cdot 31^{4} + 29\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 23 a + 28 + \left(30 a + 30\right)\cdot 31 + \left(27 a + 3\right)\cdot 31^{2} + \left(12 a + 14\right)\cdot 31^{3} + \left(11 a + 28\right)\cdot 31^{4} + \left(15 a + 21\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 21 a + 13 + \left(21 a + 29\right)\cdot 31 + \left(23 a + 1\right)\cdot 31^{2} + \left(18 a + 16\right)\cdot 31^{3} + \left(22 a + 23\right)\cdot 31^{4} + \left(16 a + 21\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$

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

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

Character values on conjugacy classes

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