Properties

Label 2.2e4_3e2_17.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{4} \cdot 3^{2} \cdot 17 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2448= 2^{4} \cdot 3^{2} \cdot 17 $
Artin number field: Splitting field of $f= x^{8} + 12 x^{6} - 6 x^{4} - 48 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 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 49 a + 5 + \left(50 a + 58\right)\cdot 59 + \left(56 a + 55\right)\cdot 59^{2} + \left(6 a + 24\right)\cdot 59^{3} + \left(15 a + 25\right)\cdot 59^{4} + \left(6 a + 4\right)\cdot 59^{5} + \left(58 a + 33\right)\cdot 59^{6} + \left(42 a + 7\right)\cdot 59^{7} + \left(33 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 22 a + 18 + \left(14 a + 21\right)\cdot 59 + \left(52 a + 51\right)\cdot 59^{2} + \left(7 a + 6\right)\cdot 59^{3} + \left(50 a + 29\right)\cdot 59^{4} + \left(7 a + 41\right)\cdot 59^{5} + \left(52 a + 56\right)\cdot 59^{6} + \left(6 a + 29\right)\cdot 59^{7} + \left(14 a + 36\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 22 a + 19 + \left(14 a + 45\right)\cdot 59 + \left(52 a + 28\right)\cdot 59^{2} + \left(7 a + 37\right)\cdot 59^{3} + \left(50 a + 46\right)\cdot 59^{4} + 7 a\cdot 59^{5} + \left(52 a + 17\right)\cdot 59^{6} + \left(6 a + 15\right)\cdot 59^{7} + \left(14 a + 15\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 23 + 19\cdot 59 + 49\cdot 59^{2} + 22\cdot 59^{3} + 30\cdot 59^{4} + 32\cdot 59^{5} + 10\cdot 59^{6} + 52\cdot 59^{7} + 23\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 54 + 8 a\cdot 59 + \left(2 a + 3\right)\cdot 59^{2} + \left(52 a + 34\right)\cdot 59^{3} + \left(43 a + 33\right)\cdot 59^{4} + \left(52 a + 54\right)\cdot 59^{5} + 25\cdot 59^{6} + \left(16 a + 51\right)\cdot 59^{7} + \left(25 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 37 a + 41 + \left(44 a + 37\right)\cdot 59 + \left(6 a + 7\right)\cdot 59^{2} + \left(51 a + 52\right)\cdot 59^{3} + \left(8 a + 29\right)\cdot 59^{4} + \left(51 a + 17\right)\cdot 59^{5} + \left(6 a + 2\right)\cdot 59^{6} + \left(52 a + 29\right)\cdot 59^{7} + \left(44 a + 22\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 7 }$ $=$ $ 37 a + 40 + \left(44 a + 13\right)\cdot 59 + \left(6 a + 30\right)\cdot 59^{2} + \left(51 a + 21\right)\cdot 59^{3} + \left(8 a + 12\right)\cdot 59^{4} + \left(51 a + 58\right)\cdot 59^{5} + \left(6 a + 41\right)\cdot 59^{6} + \left(52 a + 43\right)\cdot 59^{7} + \left(44 a + 43\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 8 }$ $=$ $ 36 + 39\cdot 59 + 9\cdot 59^{2} + 36\cdot 59^{3} + 28\cdot 59^{4} + 26\cdot 59^{5} + 48\cdot 59^{6} + 6\cdot 59^{7} + 35\cdot 59^{8} +O\left(59^{ 9 }\right)$

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

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

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