Properties

Label 4.11e2_233e2.8t23.4c1
Dimension 4
Group $\textrm{GL(2,3)}$
Conductor $ 11^{2} \cdot 233^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$\textrm{GL(2,3)}$
Conductor:$6568969= 11^{2} \cdot 233^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 3 x^{6} + x^{5} + 8 x^{4} + 2 x^{3} - 17 x^{2} - 49 x + 41 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\textrm{GL(2,3)}$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 55 a + 64 + \left(30 a + 8\right)\cdot 67 + \left(4 a + 57\right)\cdot 67^{2} + \left(13 a + 12\right)\cdot 67^{3} + \left(11 a + 22\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 64 a + 54 + \left(51 a + 43\right)\cdot 67 + \left(11 a + 30\right)\cdot 67^{2} + \left(18 a + 53\right)\cdot 67^{3} + \left(38 a + 11\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 + 49\cdot 67 + 43\cdot 67^{2} + 22\cdot 67^{3} + 3\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 42 + \left(15 a + 53\right)\cdot 67 + \left(55 a + 25\right)\cdot 67^{2} + \left(48 a + 47\right)\cdot 67^{3} + \left(28 a + 12\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 51 a + 25 + \left(58 a + 54\right)\cdot 67 + \left(63 a + 5\right)\cdot 67^{2} + \left(19 a + 42\right)\cdot 67^{3} + \left(28 a + 9\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 15 + 10\cdot 67 + 59\cdot 67^{2} + 37\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 12 a + 16 + \left(36 a + 10\right)\cdot 67 + \left(62 a + 44\right)\cdot 67^{2} + \left(53 a + 60\right)\cdot 67^{3} + \left(55 a + 53\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 16 a + 28 + \left(8 a + 37\right)\cdot 67 + \left(3 a + 1\right)\cdot 67^{2} + \left(47 a + 58\right)\cdot 67^{3} + \left(38 a + 35\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,7)(2,5)(3,6)(4,8)$$-4$
$12$$2$$(1,8)(3,6)(4,7)$$0$
$8$$3$$(2,4,3)(5,8,6)$$1$
$6$$4$$(1,3,7,6)(2,4,5,8)$$0$
$8$$6$$(1,7)(2,6,4,5,3,8)$$-1$
$6$$8$$(1,2,4,6,7,5,8,3)$$0$
$6$$8$$(1,5,4,3,7,2,8,6)$$0$
The blue line marks the conjugacy class containing complex conjugation.