Properties

Label 2.2e5_59.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{5} \cdot 59 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1888= 2^{5} \cdot 59 $
Artin number field: Splitting field of $f= x^{8} + 8 x^{6} + 14 x^{4} - 24 x^{2} - 59 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.59.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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