Properties

Label 2.1879.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 1879 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1879 $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 4 x^{6} - 7 x^{5} - 9 x^{4} - 29 x^{3} - 40 x^{2} - 35 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.1879.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 20 a + 2 + \left(30 a + 8\right)\cdot 37 + 19 a\cdot 37^{2} + \left(35 a + 6\right)\cdot 37^{3} + \left(9 a + 36\right)\cdot 37^{4} + \left(11 a + 34\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 32 + 34\cdot 37 + 34\cdot 37^{2} + 8\cdot 37^{3} + 24\cdot 37^{4} + 7\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 15 a + 35 + \left(10 a + 25\right)\cdot 37 + \left(16 a + 35\right)\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(9 a + 22\right)\cdot 37^{4} + \left(22 a + 25\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 18 a + 10 + \left(17 a + 16\right)\cdot 37 + \left(3 a + 25\right)\cdot 37^{2} + \left(30 a + 25\right)\cdot 37^{3} + \left(18 a + 33\right)\cdot 37^{4} + \left(21 a + 27\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 33 + 16\cdot 37 + 37^{2} + 9\cdot 37^{3} + 13\cdot 37^{4} + 3\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 19 a + 8 + \left(19 a + 31\right)\cdot 37 + \left(33 a + 21\right)\cdot 37^{2} + \left(6 a + 31\right)\cdot 37^{3} + \left(18 a + 4\right)\cdot 37^{4} + \left(15 a + 21\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 22 a + 21 + \left(26 a + 15\right)\cdot 37 + \left(20 a + 16\right)\cdot 37^{2} + \left(24 a + 4\right)\cdot 37^{3} + \left(27 a + 10\right)\cdot 37^{4} + \left(14 a + 31\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 17 a + 8 + \left(6 a + 36\right)\cdot 37 + \left(17 a + 11\right)\cdot 37^{2} + \left(a + 17\right)\cdot 37^{3} + \left(27 a + 3\right)\cdot 37^{4} + \left(25 a + 33\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$

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

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

Character values on conjugacy classes

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