Properties

Label 2.2e7_19.24t22.1c1
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} - 12 x^{4} - 16 x^{2} - 76 $ 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 $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 33 a + 10 + \left(49 a + 29\right)\cdot 59 + \left(35 a + 35\right)\cdot 59^{2} + \left(23 a + 24\right)\cdot 59^{3} + \left(29 a + 10\right)\cdot 59^{4} + \left(4 a + 4\right)\cdot 59^{5} + \left(18 a + 5\right)\cdot 59^{6} + \left(17 a + 56\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 22 a + 48 + \left(4 a + 8\right)\cdot 59 + \left(38 a + 42\right)\cdot 59^{2} + \left(22 a + 7\right)\cdot 59^{3} + 22 a\cdot 59^{4} + \left(41 a + 20\right)\cdot 59^{5} + \left(35 a + 32\right)\cdot 59^{6} + \left(24 a + 5\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 56 + 8\cdot 59 + 58\cdot 59^{2} + 51\cdot 59^{3} + 25\cdot 59^{4} + 54\cdot 59^{5} + 23\cdot 59^{6} + 36\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 33 a + 16 + \left(49 a + 13\right)\cdot 59 + \left(35 a + 37\right)\cdot 59^{2} + \left(23 a + 46\right)\cdot 59^{3} + \left(29 a + 42\right)\cdot 59^{4} + \left(4 a + 20\right)\cdot 59^{5} + \left(18 a + 40\right)\cdot 59^{6} + \left(17 a + 3\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 26 a + 49 + \left(9 a + 29\right)\cdot 59 + \left(23 a + 23\right)\cdot 59^{2} + \left(35 a + 34\right)\cdot 59^{3} + \left(29 a + 48\right)\cdot 59^{4} + \left(54 a + 54\right)\cdot 59^{5} + \left(40 a + 53\right)\cdot 59^{6} + \left(41 a + 2\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 37 a + 11 + \left(54 a + 50\right)\cdot 59 + \left(20 a + 16\right)\cdot 59^{2} + \left(36 a + 51\right)\cdot 59^{3} + \left(36 a + 58\right)\cdot 59^{4} + \left(17 a + 38\right)\cdot 59^{5} + \left(23 a + 26\right)\cdot 59^{6} + \left(34 a + 53\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 3 + 50\cdot 59 + 7\cdot 59^{3} + 33\cdot 59^{4} + 4\cdot 59^{5} + 35\cdot 59^{6} + 22\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 26 a + 43 + \left(9 a + 45\right)\cdot 59 + \left(23 a + 21\right)\cdot 59^{2} + \left(35 a + 12\right)\cdot 59^{3} + \left(29 a + 16\right)\cdot 59^{4} + \left(54 a + 38\right)\cdot 59^{5} + \left(40 a + 18\right)\cdot 59^{6} + \left(41 a + 55\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$

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

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

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$$2$$(1,8)(2,6)(4,5)$$0$
$8$$3$$(1,8,7)(3,5,4)$$-1$
$6$$4$$(1,3,5,7)(2,8,6,4)$$0$
$8$$6$$(1,3,8,5,7,4)(2,6)$$1$
$6$$8$$(1,3,4,6,5,7,8,2)$$-\zeta_{8}^{3} - \zeta_{8}$
$6$$8$$(1,7,4,2,5,3,8,6)$$\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.