Properties

Label 2.2e7_19.24t22.5c2
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:
$r_{ 1 }$ $=$ $ 5 a + 10 + \left(10 a + 7\right)\cdot 13 + \left(6 a + 8\right)\cdot 13^{2} + \left(6 a + 6\right)\cdot 13^{3} + 8 a\cdot 13^{4} + \left(a + 8\right)\cdot 13^{5} + \left(11 a + 4\right)\cdot 13^{6} + \left(3 a + 7\right)\cdot 13^{7} + \left(5 a + 2\right)\cdot 13^{8} + \left(9 a + 11\right)\cdot 13^{9} + \left(7 a + 9\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 3 + 6\cdot 13 + 8\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 9\cdot 13^{5} + 2\cdot 13^{6} + 5\cdot 13^{7} + 5\cdot 13^{8} + 12\cdot 13^{9} + 9\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 9 + \left(5 a + 7\right)\cdot 13 + 2\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(5 a + 5\right)\cdot 13^{4} + \left(5 a + 6\right)\cdot 13^{5} + \left(12 a + 9\right)\cdot 13^{6} + \left(5 a + 9\right)\cdot 13^{7} + \left(10 a + 10\right)\cdot 13^{8} + \left(10 a + 12\right)\cdot 13^{9} + \left(7 a + 7\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 8 a + 2 + 2 a\cdot 13 + \left(6 a + 5\right)\cdot 13^{2} + \left(6 a + 6\right)\cdot 13^{3} + \left(4 a + 2\right)\cdot 13^{4} + \left(11 a + 1\right)\cdot 13^{5} + \left(a + 1\right)\cdot 13^{6} + 9 a\cdot 13^{7} + \left(7 a + 4\right)\cdot 13^{8} + \left(3 a + 2\right)\cdot 13^{9} + \left(5 a + 8\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 3 + \left(2 a + 5\right)\cdot 13 + \left(6 a + 4\right)\cdot 13^{2} + \left(6 a + 6\right)\cdot 13^{3} + \left(4 a + 12\right)\cdot 13^{4} + \left(11 a + 4\right)\cdot 13^{5} + \left(a + 8\right)\cdot 13^{6} + \left(9 a + 5\right)\cdot 13^{7} + \left(7 a + 10\right)\cdot 13^{8} + \left(3 a + 1\right)\cdot 13^{9} + \left(5 a + 3\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 10 + 6\cdot 13 + 4\cdot 13^{2} + 11\cdot 13^{3} + 7\cdot 13^{4} + 3\cdot 13^{5} + 10\cdot 13^{6} + 7\cdot 13^{7} + 7\cdot 13^{8} + 3\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 5 a + 4 + \left(7 a + 5\right)\cdot 13 + \left(12 a + 10\right)\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + \left(7 a + 7\right)\cdot 13^{4} + \left(7 a + 6\right)\cdot 13^{5} + 3\cdot 13^{6} + \left(7 a + 3\right)\cdot 13^{7} + \left(2 a + 2\right)\cdot 13^{8} + 2 a\cdot 13^{9} + \left(5 a + 5\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 5 a + 11 + \left(10 a + 12\right)\cdot 13 + \left(6 a + 7\right)\cdot 13^{2} + \left(6 a + 6\right)\cdot 13^{3} + \left(8 a + 10\right)\cdot 13^{4} + \left(a + 11\right)\cdot 13^{5} + \left(11 a + 11\right)\cdot 13^{6} + \left(3 a + 12\right)\cdot 13^{7} + \left(5 a + 8\right)\cdot 13^{8} + \left(9 a + 10\right)\cdot 13^{9} + \left(7 a + 4\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$

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

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

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