Properties

Label 2.7_157.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 7 \cdot 157 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1099= 7 \cdot 157 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 2 x^{6} - 3 x^{4} + 11 x^{3} - 13 x^{2} + 15 x - 11 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.7_157.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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