Properties

Label 2.37_59.24t22.3c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 37 \cdot 59 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2183= 37 \cdot 59 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 5 x^{6} - x^{5} + 12 x^{4} + 18 x^{3} - 15 x^{2} - 27 x + 4 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.37_59.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 36 + 57\cdot 79 + 33\cdot 79^{2} + 40\cdot 79^{3} + 42\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 51 + \left(a + 2\right)\cdot 79 + \left(7 a + 39\right)\cdot 79^{2} + \left(a + 38\right)\cdot 79^{3} + \left(12 a + 52\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 26 + 45\cdot 79 + 27\cdot 79^{2} + 50\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 66 a + 35 + \left(77 a + 27\right)\cdot 79 + \left(34 a + 4\right)\cdot 79^{2} + \left(47 a + 50\right)\cdot 79^{3} + \left(76 a + 13\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 54 a + 59 + \left(55 a + 70\right)\cdot 79 + \left(73 a + 53\right)\cdot 79^{2} + \left(33 a + 40\right)\cdot 79^{3} + \left(41 a + 57\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 76 a + 54 + 77 a\cdot 79 + \left(71 a + 45\right)\cdot 79^{2} + \left(77 a + 32\right)\cdot 79^{3} + \left(66 a + 63\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 13 a + 22 + \left(a + 39\right)\cdot 79 + \left(44 a + 40\right)\cdot 79^{2} + \left(31 a + 62\right)\cdot 79^{3} + \left(2 a + 42\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 25 a + 34 + \left(23 a + 72\right)\cdot 79 + \left(5 a + 71\right)\cdot 79^{2} + 45 a\cdot 79^{3} + \left(37 a + 65\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

Character values on conjugacy classes

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