Properties

Label 2.2e3_3e2_17.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 3^{2} \cdot 17 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1224= 2^{3} \cdot 3^{2} \cdot 17 $
Artin number field: Splitting field of $f= x^{8} - 6 x^{4} - 24 x^{2} - 51 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3_17.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 40 + 49\cdot 59 + 13\cdot 59^{2} + 5\cdot 59^{3} + 28\cdot 59^{4} + 54\cdot 59^{5} + 16\cdot 59^{6} + 26\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 8 a + 55 + \left(52 a + 36\right)\cdot 59 + \left(34 a + 8\right)\cdot 59^{2} + \left(4 a + 15\right)\cdot 59^{3} + \left(46 a + 38\right)\cdot 59^{4} + \left(11 a + 46\right)\cdot 59^{5} + \left(30 a + 49\right)\cdot 59^{6} + \left(45 a + 21\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 24 + \left(8 a + 33\right)\cdot 59 + \left(28 a + 17\right)\cdot 59^{2} + \left(40 a + 44\right)\cdot 59^{3} + \left(38 a + 26\right)\cdot 59^{4} + \left(29 a + 48\right)\cdot 59^{5} + \left(a + 53\right)\cdot 59^{6} + \left(7 a + 42\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 8 a + 27 + \left(8 a + 25\right)\cdot 59 + \left(28 a + 21\right)\cdot 59^{2} + \left(40 a + 2\right)\cdot 59^{3} + \left(38 a + 34\right)\cdot 59^{4} + \left(29 a + 19\right)\cdot 59^{5} + \left(a + 33\right)\cdot 59^{6} + \left(7 a + 10\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 19 + 9\cdot 59 + 45\cdot 59^{2} + 53\cdot 59^{3} + 30\cdot 59^{4} + 4\cdot 59^{5} + 42\cdot 59^{6} + 32\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 51 a + 4 + \left(6 a + 22\right)\cdot 59 + \left(24 a + 50\right)\cdot 59^{2} + \left(54 a + 43\right)\cdot 59^{3} + \left(12 a + 20\right)\cdot 59^{4} + \left(47 a + 12\right)\cdot 59^{5} + \left(28 a + 9\right)\cdot 59^{6} + \left(13 a + 37\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 51 a + 35 + \left(50 a + 25\right)\cdot 59 + \left(30 a + 41\right)\cdot 59^{2} + \left(18 a + 14\right)\cdot 59^{3} + \left(20 a + 32\right)\cdot 59^{4} + \left(29 a + 10\right)\cdot 59^{5} + \left(57 a + 5\right)\cdot 59^{6} + \left(51 a + 16\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 51 a + 32 + \left(50 a + 33\right)\cdot 59 + \left(30 a + 37\right)\cdot 59^{2} + \left(18 a + 56\right)\cdot 59^{3} + \left(20 a + 24\right)\cdot 59^{4} + \left(29 a + 39\right)\cdot 59^{5} + \left(57 a + 25\right)\cdot 59^{6} + \left(51 a + 48\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$

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

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

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