Properties

Label 2.2e7_19.24t22.4c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{7} \cdot 19 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2432= 2^{7} \cdot 19 $
Artin number field: Splitting field of $f= x^{8} - 8 x^{6} + 22 x^{4} - 16 x^{2} - 19 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 12 a + 11 + \left(11 a + 10\right)\cdot 13 + 12 a\cdot 13^{2} + \left(11 a + 6\right)\cdot 13^{3} + \left(2 a + 3\right)\cdot 13^{4} + \left(4 a + 8\right)\cdot 13^{5} + \left(7 a + 12\right)\cdot 13^{6} + \left(10 a + 8\right)\cdot 13^{7} + 4\cdot 13^{8} + \left(3 a + 5\right)\cdot 13^{9} + \left(7 a + 10\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 2 + 7\cdot 13 + 11\cdot 13^{2} + 4\cdot 13^{3} + 5\cdot 13^{4} + 3\cdot 13^{5} + 12\cdot 13^{6} + 13^{7} + 10\cdot 13^{8} + 3\cdot 13^{9} + 5\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 3 }$ $=$ $ a + 6 + \left(3 a + 5\right)\cdot 13 + \left(12 a + 8\right)\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + \left(3 a + 9\right)\cdot 13^{4} + \left(7 a + 4\right)\cdot 13^{5} + \left(2 a + 2\right)\cdot 13^{6} + \left(3 a + 6\right)\cdot 13^{7} + \left(a + 7\right)\cdot 13^{8} + \left(4 a + 11\right)\cdot 13^{9} + \left(8 a + 10\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 4 }$ $=$ $ a + 10 + \left(a + 10\right)\cdot 13 + 13^{2} + \left(a + 5\right)\cdot 13^{3} + \left(10 a + 7\right)\cdot 13^{4} + \left(8 a + 9\right)\cdot 13^{5} + \left(5 a + 2\right)\cdot 13^{6} + \left(2 a + 12\right)\cdot 13^{7} + \left(12 a + 7\right)\cdot 13^{8} + \left(9 a + 7\right)\cdot 13^{9} + \left(5 a + 1\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 5 }$ $=$ $ a + 2 + \left(a + 2\right)\cdot 13 + 12\cdot 13^{2} + \left(a + 6\right)\cdot 13^{3} + \left(10 a + 9\right)\cdot 13^{4} + \left(8 a + 4\right)\cdot 13^{5} + 5 a\cdot 13^{6} + \left(2 a + 4\right)\cdot 13^{7} + \left(12 a + 8\right)\cdot 13^{8} + \left(9 a + 7\right)\cdot 13^{9} + \left(5 a + 2\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 11 + 5\cdot 13 + 13^{2} + 8\cdot 13^{3} + 7\cdot 13^{4} + 9\cdot 13^{5} + 11\cdot 13^{7} + 2\cdot 13^{8} + 9\cdot 13^{9} + 7\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 12 a + 7 + \left(9 a + 7\right)\cdot 13 + 4\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(9 a + 3\right)\cdot 13^{4} + \left(5 a + 8\right)\cdot 13^{5} + \left(10 a + 10\right)\cdot 13^{6} + \left(9 a + 6\right)\cdot 13^{7} + \left(11 a + 5\right)\cdot 13^{8} + \left(8 a + 1\right)\cdot 13^{9} + \left(4 a + 2\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 12 a + 3 + \left(11 a + 2\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(11 a + 7\right)\cdot 13^{3} + \left(2 a + 5\right)\cdot 13^{4} + \left(4 a + 3\right)\cdot 13^{5} + \left(7 a + 10\right)\cdot 13^{6} + 10 a\cdot 13^{7} + 5\cdot 13^{8} + \left(3 a + 5\right)\cdot 13^{9} + \left(7 a + 11\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$

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

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

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$$(1,7,8)(3,4,5)$$-1$
$6$$4$$(1,7,5,3)(2,4,6,8)$$0$
$8$$6$$(1,5)(2,4,3,6,8,7)$$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.