Properties

Label 2.2e4_3e2_17.24t22.3c2
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

Learn more about

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} + 42 x^{4} + 72 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:
$r_{ 1 }$ $=$ $ 52 a + 38 + \left(7 a + 56\right)\cdot 59 + \left(a + 38\right)\cdot 59^{2} + \left(44 a + 53\right)\cdot 59^{3} + \left(27 a + 48\right)\cdot 59^{4} + \left(54 a + 26\right)\cdot 59^{5} + 15\cdot 59^{6} + \left(9 a + 3\right)\cdot 59^{7} + \left(22 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 52 a + 28 + \left(7 a + 46\right)\cdot 59 + \left(a + 26\right)\cdot 59^{2} + \left(44 a + 21\right)\cdot 59^{3} + \left(27 a + 26\right)\cdot 59^{4} + \left(54 a + 5\right)\cdot 59^{5} + 38\cdot 59^{6} + \left(9 a + 47\right)\cdot 59^{7} + \left(22 a + 21\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 56 + \left(36 a + 43\right)\cdot 59 + \left(8 a + 13\right)\cdot 59^{2} + \left(23 a + 22\right)\cdot 59^{3} + \left(58 a + 41\right)\cdot 59^{4} + \left(4 a + 26\right)\cdot 59^{5} + \left(6 a + 58\right)\cdot 59^{6} + \left(17 a + 23\right)\cdot 59^{7} + \left(27 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 7 + 47\cdot 59 + 17\cdot 59^{2} + 46\cdot 59^{3} + 21\cdot 59^{4} + 22\cdot 59^{5} + 22\cdot 59^{6} + 21\cdot 59^{7} + 30\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 21 + \left(51 a + 2\right)\cdot 59 + \left(57 a + 20\right)\cdot 59^{2} + \left(14 a + 5\right)\cdot 59^{3} + \left(31 a + 10\right)\cdot 59^{4} + \left(4 a + 32\right)\cdot 59^{5} + \left(58 a + 43\right)\cdot 59^{6} + \left(49 a + 55\right)\cdot 59^{7} + \left(36 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 31 + \left(51 a + 12\right)\cdot 59 + \left(57 a + 32\right)\cdot 59^{2} + \left(14 a + 37\right)\cdot 59^{3} + \left(31 a + 32\right)\cdot 59^{4} + \left(4 a + 53\right)\cdot 59^{5} + \left(58 a + 20\right)\cdot 59^{6} + \left(49 a + 11\right)\cdot 59^{7} + \left(36 a + 37\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 7 }$ $=$ $ 53 a + 3 + \left(22 a + 15\right)\cdot 59 + \left(50 a + 45\right)\cdot 59^{2} + \left(35 a + 36\right)\cdot 59^{3} + 17\cdot 59^{4} + \left(54 a + 32\right)\cdot 59^{5} + 52 a\cdot 59^{6} + \left(41 a + 35\right)\cdot 59^{7} + \left(31 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 8 }$ $=$ $ 52 + 11\cdot 59 + 41\cdot 59^{2} + 12\cdot 59^{3} + 37\cdot 59^{4} + 36\cdot 59^{5} + 36\cdot 59^{6} + 37\cdot 59^{7} + 28\cdot 59^{8} +O\left(59^{ 9 }\right)$

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

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

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