Properties

Label 2.3e2_109.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{2} \cdot 109 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$981= 3^{2} \cdot 109 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 10 x^{4} + 2 x^{3} + 10 x^{2} - 13 x - 8 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3_109.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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