Properties

Label 2.2e3_3e2_5e2.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1800= 2^{3} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{8} - 6 x^{4} - 12 x^{2} - 3 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 15 a + 36 + \left(6 a + 6\right)\cdot 41 + \left(7 a + 12\right)\cdot 41^{2} + \left(15 a + 26\right)\cdot 41^{3} + \left(37 a + 26\right)\cdot 41^{4} + \left(40 a + 12\right)\cdot 41^{5} + \left(10 a + 1\right)\cdot 41^{6} + \left(39 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 15 a + 1 + \left(6 a + 30\right)\cdot 41 + \left(7 a + 13\right)\cdot 41^{2} + \left(15 a + 17\right)\cdot 41^{3} + \left(37 a + 40\right)\cdot 41^{4} + \left(40 a + 24\right)\cdot 41^{5} + \left(10 a + 6\right)\cdot 41^{6} + \left(39 a + 40\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 24 a + 5 + \left(14 a + 31\right)\cdot 41 + \left(36 a + 34\right)\cdot 41^{2} + \left(35 a + 25\right)\cdot 41^{3} + \left(7 a + 26\right)\cdot 41^{4} + \left(28 a + 2\right)\cdot 41^{5} + \left(25 a + 37\right)\cdot 41^{6} + \left(2 a + 8\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 18 + 7\cdot 41 + 31\cdot 41^{2} + 19\cdot 41^{3} + 25\cdot 41^{4} + 18\cdot 41^{5} + 7\cdot 41^{6} + 12\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 26 a + 5 + \left(34 a + 34\right)\cdot 41 + \left(33 a + 28\right)\cdot 41^{2} + \left(25 a + 14\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} + 28\cdot 41^{5} + \left(30 a + 39\right)\cdot 41^{6} + \left(a + 23\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 26 a + 40 + \left(34 a + 10\right)\cdot 41 + \left(33 a + 27\right)\cdot 41^{2} + \left(25 a + 23\right)\cdot 41^{3} + 3 a\cdot 41^{4} + 16\cdot 41^{5} + \left(30 a + 34\right)\cdot 41^{6} + a\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 17 a + 36 + \left(26 a + 9\right)\cdot 41 + \left(4 a + 6\right)\cdot 41^{2} + \left(5 a + 15\right)\cdot 41^{3} + \left(33 a + 14\right)\cdot 41^{4} + \left(12 a + 38\right)\cdot 41^{5} + \left(15 a + 3\right)\cdot 41^{6} + \left(38 a + 32\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 23 + 33\cdot 41 + 9\cdot 41^{2} + 21\cdot 41^{3} + 15\cdot 41^{4} + 22\cdot 41^{5} + 33\cdot 41^{6} + 28\cdot 41^{7} +O\left(41^{ 8 }\right)$

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

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

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