Properties

Label 2.3600.24t22.a.a
Dimension $2$
Group $\textrm{GL(2,3)}$
Conductor $3600$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $\textrm{GL(2,3)}$
Conductor: \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.2.5598720000.2
Galois orbit size: $2$
Smallest permutation container: 24T22
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.10800.2

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} - 2x^{6} + 4x^{5} + 4x^{4} - 2x^{3} - 32x^{2} + 16x - 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 28 a + 8 + \left(2 a + 21\right)\cdot 41 + \left(13 a + 13\right)\cdot 41^{2} + \left(a + 14\right)\cdot 41^{3} + \left(11 a + 40\right)\cdot 41^{4} + \left(5 a + 25\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + 15 + \left(36 a + 7\right)\cdot 41 + \left(24 a + 33\right)\cdot 41^{2} + 11 a\cdot 41^{3} + \left(33 a + 27\right)\cdot 41^{4} + \left(34 a + 19\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 10 + \left(30 a + 38\right)\cdot 41 + \left(18 a + 35\right)\cdot 41^{2} + \left(13 a + 29\right)\cdot 41^{3} + \left(5 a + 38\right)\cdot 41^{4} + \left(34 a + 18\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + 10 + \left(38 a + 1\right)\cdot 41 + \left(27 a + 9\right)\cdot 41^{2} + \left(39 a + 5\right)\cdot 41^{3} + \left(29 a + 31\right)\cdot 41^{4} + \left(35 a + 30\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 33 a + 39 + \left(4 a + 25\right)\cdot 41 + \left(16 a + 30\right)\cdot 41^{2} + \left(29 a + 10\right)\cdot 41^{3} + \left(7 a + 33\right)\cdot 41^{4} + \left(6 a + 8\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 16 + 14\cdot 41 + 18\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} + 36\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 31 + 17\cdot 41 + 2\cdot 41^{2} + 13\cdot 41^{3} + 2\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 32 a + 37 + \left(10 a + 37\right)\cdot 41 + \left(22 a + 20\right)\cdot 41^{2} + \left(27 a + 10\right)\cdot 41^{3} + 35 a\cdot 41^{4} + \left(6 a + 34\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,3)(2,5)(4,8)(6,7)$$-2$
$12$$2$$(2,5)(4,7)(6,8)$$0$
$8$$3$$(1,6,5)(2,3,7)$$-1$
$6$$4$$(1,6,3,7)(2,4,5,8)$$0$
$8$$6$$(1,2,6,3,5,7)(4,8)$$1$
$6$$8$$(1,6,2,8,3,7,5,4)$$-\zeta_{8}^{3} - \zeta_{8}$
$6$$8$$(1,7,2,4,3,6,5,8)$$\zeta_{8}^{3} + \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.